bessel functions (tables of some indefinite integrals)

Werner Rosenheinrich 31.01.2012UNIVERSITY OF APPLIED SCIENCES JENA First variant: 24.09.2003Jena, Germany

TABLES OF SOME INDEFINITE INTEGRALS OF BESSEL FUNCTIONS

Integrals of the type ∫xJ2

0 (x) dx or∫

xJ0(ax)J0(bx) dx

are well-known.Most of the following integrals are not found in the widely used tables of Gradstein/Ryshik, Bate-man/Erdélyi, Abramowitz/ Stegun or Jahnke/Emde/Lösch. The goal of this table was to expressthe integrals by Bessel and Struve functions. Indeed, there occured some exceptions. Generally, integralsof the type

∫xµJν(x) dx may be written with Lommel functions, see [8], 10 -74.

Reccurence formulas are included for some cases in order to obtain more integrals.

Partially the integrals may be found by MAPLE as well. In some cases MAPLE gives results with hyper-geometric functions, see also [2], 9.6. .Some known integrals are included for completeness.

Here Zν(x) denotes some Bessel function or modified Bessel function of the first kind. Let α 6= β.

When a formula is continued in the next line, then the last sign ’+’ or ’-’ is repeated in the beginning ofthe new line.

Defined functions:

Function Φ(x) Ψ(x) Θ(x) Ω(x) Λ0(x) Λ∗0(x) Λ1(x) Λ∗1(x) Θ0, Ω0 Θ1, Ω1

Page 3 3 21 21 104 108 109 110 55 57

References:

[1] M. Abramowitz, I. Stegun: Handbook of Mathematical Functions, Dover Publications, NY, 1970[2] Y. L. Luke: Mathematical Functions and their Approximations, Academic Press, NY, 1975[3] Y. L. Luke: Integrals of Bessel Functions, MacGraw-Hill, NY, 1962[4] A. P. Prudnikov, . A. Bryqkov, O. I. Mariqev:

Integraly i rdy - Special~nye funkcii, Nauka, Moskva, 1983[5] E. Jahnke, F. Emde, F. Lösch: Tafeln höherer Funktionen, 6. Auflage, B. G. Teubner, Stuttgart, 1960[6] I. S. Gradstein, I. M. Ryshik: Summen-, Produkt- und Integraltafeln / Tables of Series, Products, and

Integrals, Band 1 / Volume 1, Verlag Harri Deutsch, Thun · Frankfurt/M, 1981[7] I. S. Gradstein, I. M. Ryshik: Summen-, Produkt- und Integraltafeln / Tables of Series, Products, and

Integrals, Band 2 / Volume 2, Verlag Harri Deutsch, Thun · Frankfurt/M, 1981[8] G. N. Watson: A Treatise on the Theory of Bessel Functions, Cambridge, University Press, 1922 / 1995[9] P. Humbert: Bessel-integral functions, Proceedings of the Edinburgh Mathematical Society (Series 2),

1933, 3:276-285

I wish to express my thanks to B. Eckstein and S. O. Zafra for their remarks.

1

Type Integrand n PageI x2nZ0(x) −6 ≤ n ≤ 6 3

II x2n+1Z0(x) 0 ≤ n ≤ 7 5III x−2n−1Z0(x) 1 ≤ n ≤ 6 6IV x2nZ1(x) 0 ≤ n ≤ 7 7V x−2nZ1(x) 1 ≤ n ≤ 6 8

VI x2n+1Z1(x) −6 ≤ n ≤ 5 9VII x2n+1Z 2

ν (x) −1/0 ≤ n ≤ 7 11VIII x−2nZ 2

ν (x) 1 ≤ n ≤ 5 15IX x2nZ0(x)Z1(x) 0 ≤ n ≤ 8 17X x−(2n+1)Z0(x)Z1(x) 0 ≤ n ≤ 4 19

XI x2n Z 2ν (x) 0 ≤ n ≤ 8 21

a) The Functions Θ(x) and Ω(x) 21b) Table of Θ(x) 26c) Integrals 35

XII x2n+1 Z0(x)Z1(x) 0 ≤ n ≤ 8 40XIII x2n+1Zν(αx)Zν(βx) 0 ≤ n ≤ 5 43XIV x2nZ0(αx)Z1(βx) 1 ≤ n ≤ 5 51XV x2nZν(αx)Zν(βx) 0 ≤ n ≤ 5 55

a) Basic Integrals 55b) Integrals 65

XVI x2n+1Z0(αx)Z1(βx) −1 ≤ n ≤ 5 72XVII x2n+1J0(x)I0(x) 0 ≤ n ≤ 7 76

XVIII x2nJ0(x)I1(x) 1 ≤ n ≤ 8 77XIX x2nJ1(x)I0(x) 1 ≤ n ≤ 8 78XX x2n+1J1(x)I1(x) 0 ≤ n ≤ 7 79

XXI x2n+1J0(αx)I0(βx) 0 ≤ n ≤ 5 80XXII x2nJ0(αx)I1(βx) 1 ≤ n ≤ 5 82

XXIII x2nJ1(αx)I0(βx) 1 ≤ n ≤ 5 84XXIV x2n+1J1(αx)I1(βx) 0 ≤ n ≤ 5 86XXV x−mZ n

0 (x)Z 4−n1 (x) 0, 1, 2 88

XXVI xn+1/2Jν(x) −13 ≤ n ≤ 14 89XXVII xn e±x Iν(x) −1 / 0 ≤ n ≤ 7 97

XXVIII xn ·

sincos

x · Jν(x) 0 ≤ n ≤ 7 99

XXIX x2n+1 lnx · Z0(x) 0 ≤ n ≤ 7 102XXX x2n lnx · Z1(x) 0 ≤ n ≤ 7 104

XXXI x2n+ν lnx · Zν(x) 106a) The Functions Λk and Λ∗k 106b) Basic Integrals 110c) Integrals of x2n lnx · Z0(x) 0 ≤ n ≤ 9 113c) Integrals of x2n+1 lnx · Z1(x) 0 ≤ n ≤ 9 115

XXXII Orthogonal Polynomials 119a) Legendre Polynomials Pn(x) 0 ≤ n ≤ 15 119b) Chebyshev Polynomials Tn(x) 0 ≤ n ≤ 15 124c) Chebyshev Polynomials Un(x) 0 ≤ n ≤ 10 129d) Laguerre Polynomials Ln(x) 0 ≤ n ≤ 12 132e) Hermite Polynomials Hn(x) 0 ≤ n ≤ 15 136

XXXIII x−1 Zν(x + α)Z1(x) - 141XXXIV Higher Antiderivatives - 142

IC Miscellaneous - 143

2

I. Integrals of the type∫

x2nZ0(x) dx

LetΦ(x) =

πx

2[J1(x) ·H0(x)− J0(x) ·H1(x)] ,

where Hν(x) denotes the Struve function, see [1], chapter 11 and 12.And let

Ψ(x) =πx

2[I0(x) · L1(x)− I1(x) · L0(x)]

be defined with the modified Struve function Lν(x).∫J0(x) dx = xJ0(x) + Φ(x) = Λ0(x)∫I0(x) dx = xI0(x) + Ψ(x) = Λ∗0(x)

The function Λ0(x) is discussed in XXXI a) on page 106 and Λ∗0(x) on page 108.∫x2 J0(x) dx = x2J1(x)− Φ(x)

∫x2 I0(x) dx = x2I1(x)−Ψ(x)∫

x4 J0(x) dx = (x4 − 9x2)J1(x) + 3x3J0(x) + 9Φ(x)∫x4 I0(x) dx = (x4 + 9x2)I1(x)− 3x3I0(x) + 9Ψ(x)∫

x6 J0(x) dx = (x6 − 25x4 + 225x2)J1(x) + (5x5 − 75x3)J0(x)− 225Φ(x)∫x6 I0(x) dx = (x6 + 25x4 + 225x2)I1(x)− (5x5 + 75x3)I0(x) + 225Ψ(x)∫

x8 J0(x) dx = (x8 − 49x6 + 1 225x4 − 11 025x2)J1(x) + (7x7 − 245x5 + 3 675x3)J0(x) + 11 025Φ(x)∫x8 I0(x) dx = (x8 + 49x6 + 1 225x4 + 11 025x2)I1(x)− (7x7 + 245x5 + 3 675x3)I0(x) + 11 025Ψ(x)∫

x10 J0(x) dx = (x10 − 81x8 + 3 969x6 − 99 225x4 + 893 025)J1(x)+

+(9x9 − 567x7 + 19 845x5 − 297 675x3)J0(x)− 893 025Φ(x)∫x10 I0(x) dx = (x10 + 81x8 + 3 969x6 + 99 225x4 + 893 025)I1(x)−

−(9x9 + 567x7 + 19 845x5 + 297 675x3)I0(x) + 893 025Ψ(x)∫x12 J0(x) dx = (11x11 − 1 089x9 + 68 607x7 − 2 401 245x5 + 36 018 675x3)J0(x)+

+(x12 − 121x10 + 9 801x8 − 480 249x6 + 12 006 225x4 − 108 056 025x2)J1(x) + 108 056 025Φ(x) =

= (11x11 − 112 · 9x9 + 112 · 92 · 7x7 − 112 · 92 · 72 · 5x5 + 112 · 92 · 72 · 52 · 3x3) J0(x)+

+(x12 − 112x10 + 112 · 92x8 − 112 · 92 · 72x6 + 112 · 92 · 72 · 52x4 + 112 · 92 · 72 · 52 · 32x2) J1(x)−

−112 · 92 · 72 · 52 · 32 Φ(x)∫x12 I0(x) dx = (x12 + 121x10 + 9 801x8 + 480 249x6 + 12 006 225x4 + 108 056 025x2)I1(x)−

−(11x11 + 1 089x9 + 68 607x7 + 2 401 245x5 + 36 018 675x3)I0(x) + 108 056 025Ψ(x)

The general case may be generated from the last but one formula.The same holds for the following three types.

3

Recurrence formulas:∫x2n+2 J0(x) dx = (2n + 1)x2n+1 J0(x) + x2n+2 J1(x)− (2n + 1)2

∫x2n J0(x) dx∫

x2n+2 I0(x) dx = −(2n + 1)x2n+1 I0(x) + x2n+2 I1(x) + (2n + 1)2∫

x2n I0(x) dx

In the case n < 0 the previous formulas give∫J0(x)

x2dx = J1(x)− x2 + 1

xJ0(x)− Φ(x)

∫I0(x)x2

dx =x2 − 1

xI0(x)− I1(x) + Ψ(x)∫

J0(x)x4

dx =19

[x4 + x2 − 3

x3J0(x)− x2 − 1

xJ1(x) + Φ(x)

]∫

I0(x)x4

dx =19

[x4 − x2 − 3

x3I0(x)− x2 + 1

xI1(x) + Ψ(x)

]∫

J0(x)x6

dx =1

225

[x4 − x2 + 9

x4J1(x)− x6 + x4 − 3x2 + 45

x5J0(x)− Φ(x)

]∫

I0(x)x6

dx =1

225

[x6 − x4 − 3x2 − 45

x5I0(x)− x4 + x2 + 9

x4I1(x) + Ψ(x)

]∫

J0(x)x8

dx =1

11 025

[x8 + x6 − 3x4 + 45x2 − 1 575

x7J0(x)− x6 − x4 + 9x2 − 225

x6J1(x) + Φ(x)

]∫

I0(x)x8

dx =1

11 025

[x8 − x6 − 3x4 − 45x2 − 1 575

x7I0(x)− x6 + x4 + 9x2 + 225

x6I1(x) + Ψ(x)

]∫

J0(x)x10

dx =1

893 025

[x8 − x6 + 9x4 − 225x2 + 11 025

x8J1(x) −

−x10 + x8 − 3x6 + 45x4 − 1 575x2 + 99 225x9

J0(x)− Φ(x)]

∫I0(x)x10

dx =1

893 025

[x10 − x8 − 3x6 − 45x4 − 1 575x2 − 99 225

x9I0(x) −

−x8 + x6 + 9x4 + 225x2 + 11 025x8

I1(x) + Ψ(x)]

∫J0(x)x12

dx =1

108 056 025

[x12 + x10 − 3x8 + 45x6 − 1 575x4 + 99 225x2 − 9 823 275

x11J0(x) −

− x10 − x8 + 9x6 − 225x4 + 11 025x2 − 893 025x10

J1(x) + Φ(x)]

∫I0(x)x12

dx =1

108 056 025

[x12 − x10 − 3x8 − 45x6 − 1 575x4 − 99 225x2 − 9 823 275

x11I0(x) −

− x10 + x8 + 9x6 + 225x4 + 11 025x2 + 893 025x10

I1(x) + Ψ(x)]

Holds 108 056 025 = (1 · 3 · 5 · 7 · 9 · 11)2 = (11! !)2.

4

II. Integrals of the type∫

x2n+1Z0(x) dx∫xJ0(x) dx = xJ1(x)∫x I0(x) dx = x I1(x)∫

x3 J0(x) dx = x[2xJ0(x) + (x2 − 4) J1(x)

]∫

x3 I0(x) dx = x[(x2 + 4) I1(x)− 2x I0(x)

]∫

x5 J0(x) dx = x[(4x3 − 32x) J0(x) + (x4 − 16x2 + 64) J1(x)

]∫

x5 I0(x) dx = x[(x4 + 16x2 + 64) I1(x)− (4x3 + 32x) I0(x)

]∫

x7 J0(x) dx = x[(6x5 − 144x3 + 1 152x) J0(x) + (x6 − 36x4 + 576x2 − 2 304)J1(x)

]∫

x7 I0(x) dx = x[(x6 + 36x4 + 576x2 + 2 304) I1(x)− (6x5 + 144x3 + 1 152x) I0(x)

]∫

x9 J0(x) dx =

= x[(8x7 − 384x5 + 9 216x3 − 73 728x) J0(x) + (x8 − 64x6 + 2 304x4 − 36 864x2 + 147 456)J1(x)

]=

= x[(8x7 − 82 · 6x5 + 82 · 62 · 4x3 − 82 · 62 · 42 · 2x) J0(x) +

+(x8 − 82x6 + 82 · 62x4 − 82 · 62 · 42x2 + 82 · 62 · 42 · 22) J1(x)]∫

x9 I0(x) dx =

= x[(x8 + 64x6 + 2 304x4 + 36 864x2 + 147 456) I1(x)− (8x7 + 384x5 + 9 216x3 + 73 728x) I0(x)

]Let∫

xmJ0(x) dx = x[Pm(x)J0(x) + Qm(x)J1(x)] and∫

xmI0(x) dx = x[Q∗m(x)I1(x)− P ∗m(x)I0(x)] ,

then holdsP11(x) = 10 x9 − 800 x7 + 38400 x5 − 921600 x3 + 7372800 xQ11(x) = x10 − 100 x8 + 6400 x6 − 230400 x4 + 3686400x2 − 14745600P ∗11(x) = −10 x9 − 800 x7 − 38400 x5 − 921600 x3 − 7372800xQ∗11(x) = x10 + 100 x8 + 6400 x6 + 230400 x4 + 3686400x2 + 14745600

P13(x) = 12 x11 − 1440 x9 + 115200 x7 − 5529600x5 + 132710400x3 − 1061683200 xQ13(x) = x12 − 144 x10 + 14400x8 − 921600 x6 + 33177600x4 − 530841600x2 + 2123366400P ∗13(x) = −12 x11 − 1440 x9 − 115200 x7 − 5529600 x5 − 132710400x3 − 1061683200 xQ∗13(x) = x12 + 144 x10 + 14400 x8 + 921600 x6 + 33177600 x4 + 530841600x2 + 2123366400

P15(x) = 14 x13− 2352 x11 + 282240x9− 22579200 x7 + 1083801600 x5− 26011238400x3 + 208089907200 xQ15(x) =x14− 196 x12 +28224x10− 2822400 x8 +180633600x6− 6502809600 x4 +104044953600x2− 416179814400P ∗15(x) = −14 x13−2352 x11−282240 x9−22579200 x7−1083801600 x5−26011238400x3−208089907200 xQ∗15(x) =x14 +196x12 +28224x10 +2822400x8 +180633600x6 +6502809600x4 +104044953600x2 +416179814400

Recurrence formulas:∫x2n+1 J0(x) dx = 2nx2n J0(x) + x2n+1 J1(x)− 4n2

∫x2n−1 J0(x) dx∫

x2n+1 I0(x) dx = −2nx2n I0(x) + x2n+1 I1(x) + 4n2

∫x2n−1 I0(x) dx

5

III. Integrals of the type∫

x−2n−1 · J0(x) dx

The basic integral∫J0(x) dx

xcan be expressed by

∫ x

0

1− J0(t)t

dt or∫ ∞

x

J0(t) dt

t= Ji0(x) ,

see [1], equation 11.1.19 and the following formulas. There are given asymptotic expansions and polynomialapproximations as well. Tables of these functions may be found by [1], [11.13] or [11.22]. The functionJi0(x) is introduced and discussed in [9].The power series in ∫

I0(x) dx

x= lnx +

∞∑k=1

1k · (k!)2

(x

2

)2k

may be computed without numerical problems.∫J0(x) dx

x3= −J0(x)

2x2+

J1(x)4x

− 14

∫J0(x) dx

x∫I0(x) dx

x3= −I0(x)

2x2− I1(x)

4x+

14

∫I0(x) dx

x∫J0(x) dx

x5=(

132x2

− 14x4

)J0(x) +

(− 1

64x+

116x3

)J1(x) +

164

∫J0(x) dx

x∫I0(x) dx

x5= −

(1

32x2+

14x4

)I0(x)−

(1

64x+

116x3

)I1(x) +

164

∫I0(x) dx

x∫J0(x) dx

x7=−x4 + 8 x2 − 192

1152 x6J0(x) +

x4 − 4 x2 + 642304 x5

J1(x)− 12304

∫J0(x) dx

x∫I0(x) dx

x7= −x4 + 8 x2 + 192

1152 x6I0(x)− x4 + 4 x2 + 64

2304 x5I1(x) +

12304

∫I0(x) dx

x∫J0(x) dx

x9=

x6 − 8 x4 + 192 x2 − 921673728 x8

J0(x) +−x6 + 4 x4 − 64 x2 + 2304

147456 x7J1(x) +

1147456

∫J0(x) dx

x∫I0(x) dx

x9= −x6 + 8 x4 + 192 x2 + 9216

73728 x8I0(x)− x6 + 4 x4 + 64 x2 + 2304

147456 x7I1(x) +

1147456

∫I0(x) dx

x∫J0(x) dx

x11=−x8 + 8 x6 − 192 x4 + 9216 x2 − 737280

7372800 x10J0(x)+

+x8 − 4 x6 + 64 x4 − 2304 x2 + 147456

14745600 x9J1(x)− 1

14745600

∫J0(x) dx

x∫I0(x) dx

x11= −x8 + 8 x6 + 192 x4 + 9216 x2 + 737280

7372800x10I0(x)−

−x8 + 4 x6 + 64 x4 + 2304 x2 + 14745614745600 x9

I1(x) +1

14745600+∫

I0(x) dx

x∫J0(x) dx

x13=

1061683200 x12x10 − 8 x8 + 192 x6 − 9216 x4 + 737280 x2 − 884736001061683200 x12

J0(x)+

+−x10 + 4 x8 − 64 x6 + 2304 x4 − 147456 x2 + 14745600

2123366400x11J1(x) +

12123366400

∫J0(x) dx

x∫I0(x) dx

x13= −1061683200 x12x10 + 8 x8 + 192 x6 + 9216 x4 + 737280 x2 + 88473600

1061683200x12I0(x)−

−x10 + 4 x8 + 64 x6 + 2304 x4 + 147456 x2 + 147456002123366400x11

I1(x) +1

2123366400

∫I0(x) dx

x

6

IV. Integrals of the type∫

x2nZ1(x) dx∫J1(x) dx = − J0(x)∫I1(x) dx = I0(x)∫

x2 J1(x) dx = x [2J1(x)− xJ0(x)]∫x2 I1(x) dx = x [x I0(x)− 2I1(x)]∫

x4 J1(x) dx = x[(4x2 − 16) J1(x)− (x3 − 8x) J0(x)

]∫

x4 I1(x) dx = x[(x3 + 8x) I0(x)− (4x2 + 16) I1(x)

]∫

x6 J1(x) dx = x[(6x4 − 96x2 + 384) J1(x)− (x5 − 24x3 + 192x) J0(x)

]∫

x6 I1(x) dx = x[(x5 + 24x3 + 192x) I0(x)− (6x4 + 96x2 + 384) I1(x)

]∫

x8 J1(x) dx = x[(8x6 − 288x4 + 4 608x2 − 18 432)J1(x)− (x7 − 48x5 + 1 152x3 − 9 216x) J0(x)

]∫

x8 I1(x) dx = x[(x7 + 48x5 + 1 152x3 + 9 216x) I0(x)− (8x6 + 288x4 + 4 608x2 + 18 432) I1(x)

]∫

x10 J1(x) dx = x[(10x8 − 640x6 + 23 040x4 − 368 640x2 + 1 474 560) J1(x)−

− (x9 − 80x7 + 3 840x5 − 92 160x3 + 737 280x) J0(x)]

=

= x[(10x8 − 10 · 82x6 + 10 · 82 · 62x4 − 10 · 82 · 62 · 42x2 + 10 · 82 · 62 · 42 · 22) J1(x)−

− (x9 − 10 · 8x7 + 10 · 82 · 6x5 − 10 · 82 · 62 · 4x3 + 10 · 82 · 62 · 42 · 2x) J0(x)]∫

x10 I1(x) dx = x[(x9 + 80x7 + 3 840x5 + 92 160x3 + 737 280x) I0(x)−

−(10x8 + 640x6 + 23 040x4 + 368 640x2 + 1 474 560) I1(x)]

Let∫xmJ1(x) dx = x[Qm(x)J1(x)− Pm(x)J0(x)] and

∫xmI1(x) dx = x[P ∗m(x)I0(x)−Q∗m(x)I1(x)] ,

then holdsP12(x) = x11 − 120 x9 + 9600 x7 − 460800 x5 + 11059200 x3 − 88473600 xQ12(x) = 12 x10 − 1200 x8 + 76800x6 − 2764800 x4 + 44236800 x2 − 176947200P ∗12 = x11 + 120 x9 + 9600 x7 + 460800 x5 + 11059200 x3 + 88473600 xQ∗12 = 12 x10 + 1200 x8 + 76800 x6 + 2764800x4 − 44236800 x2 − 176947200

P14(x) = x13 − 168 x11 + 20160x9 − 1612800 x7 + 77414400 x5 − 1857945600 x3 + 14863564800xQ14(x) = 14 x12 − 2016 x10 + 201600 x8 − 12902400 x6 + 464486400x4 − 7431782400 x2 + 29727129600P ∗14 = x13 + 168 x11 + 20160 x9 + 1612800x7 + 77414400 x5 + 1857945600 x3 + 14863564800xQ∗14 = 14x12 + 2016 x10 + 201600 x8 + 12902400 x6 + 464486400x4 + 7431782400 x2 + 29727129600

Recurrence formulas:∫x2n+2 J1(x) dx = −x2n+2 J0(x) + (2n + 2)x2n+1 J1(x)− 4n(n + 1)

∫x2n J1(x) dx∫

x2n+2 I1(x) dx = x2n+2 I0(x)− (2n + 2)x2n+1 I1(x) + 4n(n + 1)∫

x2n I1(x) dx

7

V. Integrals of the type∫

x−2n · J1(x) dx

About the integrals ∫J0(x) dx

xand

∫I0(x) dx

x

see III, page 6. ∫J1(x) dx

x2= − 1

2xJ1(x) +

12

∫J0(x) dx

x∫I1(x) dx

x2= − 1

2xI1(x) +

12

∫I0(x) dx

x∫J1(x) dx

x4= − 1

8 x2J0(x) +

x2 − 416 x3

J1(x)− 116

∫J0(x) dx

x∫I1(x) dx

x4= − 1

8 x2I0(x)− x2 + 4

16 x3I1(x) +

116

∫I0(x) dx

x∫J1(x) dx

x6=

x2 − 8192 x4

J0(x) +−x4 + 4 x2 − 64

384 x5J1(x) +

1384

∫J0(x) dx

x∫I1(x) dx

x6= −x2 + 8

192 x4I0(x)− x4 + 4 x2 + 64

384 x5I1(x) +

1384

∫I0(x) dx

x

∫J1(x) dx

x8=−x4 + 8 x2 − 192

9216 x6J0(x) +

x6 − 4 x4 + 64 x2 − 230418432 x7

J1(x)− 118432

∫J0(x) dx

x∫I1(x) dx

x8= −x4 + 8 x2 + 192

9216 x6I0(x)− x6 + 4 x4 + 64 x2 + 2304

18432 x7I1(x) +

118432

∫I0(x) dx

x∫J1(x) dx

x10=

=x6 − 8 x4 + 192 x2 − 9216

737280 x8J0(x) +

−x8 + 4 x6 − 64 x4 + 2304 x2 − 1474561474560 x9

J1(x) +1

1474560

∫J0(x) dx

x∫I1(x) dx

x10=

= −x6 + 8 x4 + 192 x2 + 9216737280 x8

I0(x)− x8 + 4 x6 + 64 x4 + 2304 x2 + 1474561474560 x9

I1(x) +1

1474560

∫I0(x) dx

x∫J1(x) dx

x12=−x8 + 8 x6 − 192 x4 + 9216 x2 − 737280

88473600 x10J0(x)+

+x10 − 4 x8 + 64 x6 − 2304 x4 + 147456 x2 − 14745600

176947200x11J1(x)− 1

176947200

∫J0(x) dx

x∫I1(x) dx

x12= −x8 + 8 x6 + 192 x4 + 9216 x2 + 737280

88473600 x10I0(x)−

−x10 + 4 x8 + 64 x6 + 2304 x4 + 147456 x2 + 14745600176947200x11

I1(x) +1

176947200

∫I0(x) dx

x

8

VI. Integrals of the type∫

x2n+1Z1(x) dx

Φ(x) and Ψ(x) are the same as in I. . ∫xJ1(x) dx = Φ(x)

∫x I1(x) dx = −Ψ(x)∫

x3 J1(x) dx = 3x2 J1(x)− x3 J0(x)− 3Φ(x)∫x3 I1(x) dx = −3x2 I1(x) + x3 I0(x)− 3Ψ(x)∫

x5 J1(x) dx = (5x4 − 45x2) J1(x)− (x5 − 15x3) J0(x) + 45Φ(x)∫x5 I1(x) dx = −(5x4 + 45x2) I1(x) + (x5 + 15x3) I0(x)− 45Ψ(x)∫

x7 J1(x) dx = (7x6 − 175x4 − 1 575x2) J1(x)− (x7 − 35x5 + 525x3) J0(x)− 1 575Φ(x)∫x7 I1(x) dx = −(7x6 + 175x4 + 1 575x2) I1(x) + (x7 + 35x5 + 525x3) I0(x)− 1 575Ψ(x)∫

x9 J1(x) dx = (9x8−441x6+11 025x4−99 225x2) J1(x)−(x9−63x7+2205x5−33 075x3) J0(x)+99 225 Φ(x) =

= (9x8 − 9 · 72x6 + 9 · 72 · 52x4 − 9 · 72 · 52 · 32x2) J1(x)− (x9 − 9 · 7x7 + 9 · 72 · 5x5 − 9 · 72 · 52 · 3x3) J0(x)+

+9 · 72 · 52 · 32 Φ(x)∫x9 I1(x) dx = −(9x8+441x6+11 025x4+99 225x2) I1(x)+(x9+63x7+2205x5+33 075x3) I0(x)−99 225Ψ(x)

Recurrence formulas:∫x2n+1 J1(x) dx = −x2n+1 J0(x) + (2n + 1)x2n J1(x)− (2n− 1)(2n + 1)

∫x2n−1 J1(x) dx

∫x2n+1 I1(x) dx = x2n+1 I0(x)− (2n + 1)x2n I1(x) + (2n− 1)(2n + 1)

∫x2n−1 I1(x) dx

From this: ∫J1(x)

xdx = x · J0(x)− J1(x) + Φ(x)∫

I1(x)x

dx = x · I0(x)− I1(x) + Ψ(x)∫J1(x)

x3dx =

13

[x2 − 1

x2J1(x)− x2 + 1

xJ0(x)− Φ(x)

]∫

I1(x)x3

dx =13

[−x2 + 1

x2I1(x) +

x2 − 1x

I0(x) + Ψ(x)]

∫J1(x)

x5dx =

145

[x4 + x2 − 3

x3J0(x)− x4 − x2 + 9

x4J1(x) + Φ(x)

]∫

I1(x)x5

dx =145

[x4 − x2 − 3

x3I0(x)− x4 + x2 + 9

x4I1(x) + Ψ(x)

]∫

J1(x)x7

dx =1

1 575

[x6 − x4 + 9x2 − 225

x6J1(x)− x6 + x4 − 3x2 + 45

x5J0(x)− Φ(x)

]

9

∫I1(x)x7

dx =1

1 575

[−x6 + x4 + 9x2 + 225

x6I1(x) +

x6 − x4 − 3x2 − 45x5

I0(x) + Ψ(x)]

∫J1(x)

x9dx =

199 225

[x8 + x6 − 3x4 + 45x2 − 1 575

x7J0(x)− x8 − x6 + 9x4 − 225x2 + 11 025

x8J1(x) + Φ(x)

]∫

I1(x)x9

dx =

199 225

[x8 − x6 − 3x4 − 45x2 − 1 575

x7I0(x)− x8 + x6 + 9x4 + 225x2 + 11 025

x8I1(x) + Ψ(x)

]∫

J1(x)x11

dx =1

9 823 275

[x10 − x8 + 9x6 − 225x4 + 11 025x2 − 893 025

x10J1(x)−

−x10 + x8 − 3x6 + 45x4 − 1 575x2 + 99 225x9

J0(x)− Φ(x)]

∫I1(x)x11

dx =1

9 823 275

[−x10 + x8 + 9x6 + 225x4 + 11 025x2 + 893 025

x10I1(x)+

+x10 − x8 − 3x6 − 45x4 − 1 575x2 − 99 225

x9I0(x) + Ψ(x)

]Holds 9 823 275 = 1 · (3 · 5 · 7 · 9)2 · 11 = (9! !) · (11! !).

10

VII. Integrals of the type∫

x2n+1Z 2ν (x) dx∫

J21 (x)x

dx = −12[J2

0 (x) + J21 (x)

]∫

I21 (x)x

dx =12[I20 (x)− I2

1 (x)]

∫xJ2

0 (x) dx =x2

2[J2

0 (x) + J21 (x)

]∫

xJ21 (x) dx =

x

2[xJ2

0 (x) + xJ21 (x)− 2J0(x) · J1(x)

]∫

xI20 (x) dx =

x2

2[I20 (x)− I2

1 (x)]

∫xI2

1 (x) dx =x

2[xI2

1 (x)− xI20 (x) + 2I0(x) · I1(x)

]∫

x3J20 (x) dx =

x4

6J2

0 (x) +x3

3J0(x)J1(x) +

(x4

6− x2

3

)J2

1 (x)∫x3J2

1 (x) dx =x4

6J2

0 (x)− 2x3

3J0(x)J1(x) +

(x4

6+

2x2

3

)J2

1 (x)∫x3I2

0 (x) dx =x4

6I20 (x) +

x3

3I0(x)I1(x)−

(x4

6+

x2

3

)I21 (x)∫

x3I21 (x) dx = −x4

6I20 (x) +

2x3

3I0(x)I1(x) +

(x4

6− 2x2

3

)I21 (x)∫

x5J20 (x) dx =

(x6

10+

4x4

15

)J2

0 (x) +(

2x5

5− 16x3

15

)J0(x)J1(x) +

(x6

10− 8x4

15+

16x2

15

)J2

1 (x)∫x5J2

1 (x) dx =(

x6

10− 2x4

5

)J2

0 (x) +(−3x5

5+

8x3

5

)J0(x)J1(x) +

(x6

10+

4x4

5− 8x2

5

)J2

1 (x)∫x5I2

0 (x) dx =(

x6

10− 4x4

15

)I20 (x) +

(2x5

5+

16x3

15

)I0(x)I1(x)−

(x6

10+

8x4

15+

16x2

15

)I21 (x)∫

x5I21 (x) dx = −

(x6

10+

2x4

5

)I20 (x) +

(3x5

5+

8x3

5

)I0(x)I1(x) +

(x6

10− 4x4

5− 8x2

5

)I21 (x)∫

x7J20 (x) dx =

(x8

14+

18x6

35− 72x4

35

)J2

0 (x) +(

3x7

7− 108x5

35+

288x3

35

)J0(x)J1(x)+

+(

x8

14− 27x6

35+

144x4

35− 288x2

35

)J2

1 (x)∫x7J2

1 (x) dx =(

x8

14− 24x6

35+

96x4

35

)J2

0 (x) +(−4x7

7+

144x5

35− 384x3

35

)J0(x)J1(x)+

+(

x8

14+

36x6

35− 192x4

35+

384x2

35

)J2

1 (x)∫x7I2

0 (x) dx =(

x8

14− 18x6

35− 72x4

35

)I20 (x) +

(3x7

7+

108x5

35+

288x3

35

)I0(x)I1(x)−

−(

x8

14+

27x6

35+

144x4

35+

288x2

35

)I21 (x)∫

x7I21 (x) dx = −

(x8

14+

24x6

35+

96x4

35

)I20 (x) +

(4x7

7+

144x5

35+

384x3

35

)I0(x)I1(x)+

11

+(

x8

14− 36x6

35− 192x4

35− 384x2

35

)I21 (x)∫

x9J20 (x) dx =

(x10

18+

16x8

21− 256x6

35+

1024x4

35

)J2

0 (x)+

+(

4x9

9− 128x7

21+

1536x5

35− 4096x3

35

)J0(x)J1(x)+

+(

x10

18− 64x8

63+

384x6

35− 2048x4

35+

4096x2

35

)J2

1 (x)∫x9J2

1 (x) dx =(

x10

18− 20x8

21+

64x6

7− 256x4

7

)J2

0 (x)+

+(−5x9

9+

160x7

21− 384x5

7+

1024x3

7

)J0(x)J1(x)+

+(

x10

18+

80x8

63− 96x6

7+

512x4

7− 1024x2

7

)J2

1 (x)∫x9I2

0 (x) dx =(

x10

18− 16x8

21− 256x6

35− 1024x4

35

)I20 (x)+

+(

4x9

9+

128x7

21+

1536x5

35+

4096x3

35

)I0(x)I1(x)−

−(

x10

18+

64x8

63+

384x6

35+

2048x4

35+

4096x2

35

)I21 (x)∫

x9I21 (x) dx = −

(x10

18+

20x8

21+

64x6

7+

256x4

7

)I20 (x)+

+(

5x9

9+

160x7

21+

384x5

7+

1024x3

7

)I0(x)I1(x)+

+(

x10

18− 80x8

63− 96x6

7− 512x4

7− 1024x2

7

)I21 (x)

Let ∫xm · J2

0 (x) dx = Am(x) · J20 (x) + Bm(x) · J0(x) · J1(x) + Cm(x) · J2

1 (x) ,∫xm · J2

1 (x) dx = Dm(x) · J20 (x) + Em(x) · J0(x) · J1(x) + Fm(x) · J2

1 (x) ,∫xm · I2

0 (x) dx = A∗m(x) · I20 (x) + B∗m(x) · I0(x) · I1(x) + C∗m(x) · I2

1 (x) ,∫xm · I2

1 (x) dx = D∗m(x) · I2

0 (x) + E∗m(x) · I0(x) · I1(x) + F ∗m(x) · I21 (x) ,

then holdsA11 =

122

x12 +10099

x10 − 4000231

x8 +12800

77x6 − 51200

77x4

B11 =511

x11 − 100099

x9 +32000231

x7 − 7680077

x5 +204800

77x3

C11 =122

x12 − 12599

x10 +16000693

x8 − 1920077

x6 +102400

77x4 − 204800

77x2

D11 =122

x12 − 4033

x10 +160077

x8 − 1536077

x6 +61440

77x4

E11 = − 611

x11 +40033

x9 − 1280077

x7 +92160

77x5 − 245760

77x3

F11 =122

x12 +5033

x10 − 6400231

x8 +23040

77x6 − 122880

77x4 +

24576077

x2

12

A∗11 =122

x12 − 10099

x10 − 4000231

x8 − 1280077

x6 − 5120077

x4

B∗11 =511

x11 +100099

x9 +32000231

x7 +76800

77x5 +

20480077

x3

C∗11 = − 122

x12 − 12599

x10 − 16000693

x8 − 1920077

x6 − 10240077

x4 − 20480077

x2

D∗11 = − 1

22x12 − 40

33x10 − 1600

77x8 − 15360

77x6 − 61440

77x4

E∗11 =611

x11 +40033

x9 +12800

77x7 +

9216077

x5 +245760

77x3

F ∗11 =122

x12 − 5033

x10 − 6400231

x8 − 2304077

x6 − 12288077

x4 − 24576077

x2

A13 =126

x14 +180143

x12 − 4800143

x10 +5760001001

x8 − 55296001001

x6 +22118400

1001x4

B13 =613

x13 − 2160143

x11 +48000143

x9 − 46080001001

x7 +33177600

1001x5 − 88473600

1001x3

C13 =126

x14 − 216143

x12 +6000143

x10 − 7680001001

x8 +8294400

1001x6 − 44236800

1001x4 +

884736001001

x2

D13 =126

x14 − 210143

x12 +5600143

x10 − 96000143

x8 +921600

143x6 − 3686400

143x4

E13 = − 713

x13 +2520143

x11 − 56000143

x9 +768000

143x7 − 5529600

143x5 +

14745600143

x3

F13 =126

x14 +252143

x12 − 7000143

x10 +128000

143x8 − 1382400

143x6 +

7372800143

x4 − 14745600143

x2

A∗13 =126

x14 − 180143

x12 − 4800143

x10 − 5760001001

x8 − 55296001001

x6 − 221184001001

x4

B∗13 =613

x13 +2160143

x11 +48000143

x9 +4608000

1001x7 +

331776001001

x5 +88473600

1001x3

C∗13 = − 126

x14 − 216143

x12 − 6000143

x10 − 7680001001

x8 − 82944001001

x6 − 442368001001

x4 − 884736001001

x2

D∗13 = − 1

26x14 − 210

143x12 − 5600

143x10 − 96000

143x8 − 921600

143x6 − 3686400

143x4

E∗13 =713

x13 +2520143

x11 +56000143

x9 +768000

143x7 +

5529600143

x5 +14745600

143x3

F ∗13 =126

x14 − 252143

x12 − 7000143

x10 − 128000143

x8 − 1382400143

x6 − 7372800143

x4 − 14745600143

x2

A15 =130

x16 +9865

x14 − 8232143

x12 +219520

143x10 − 3763200

143x8 +

36126720143

x6 − 144506880143

x4

B15 =715

x15 − 137265

x13 +98784143

x11 − 2195200143

x9 +30105600

143x7 − 216760320

143x5 +

578027520143

x3

C15 =x16

30− 343

195x14 +

49392715

x12− 274400143

x10 +5017600

143x8− 54190080

143x6 +

289013760143

x4− 578027520143

x2

D15 =x16

30− 112

65x14 +

9408143

x12 − 250880143

x10 +4300800

143x8 − 41287680

143x6 +

165150720143

x4

E15 = − 815

x15 +156865

x13 − 112896143

x11 +2508800

143x9 − 34406400

143x7 +

247726080143

x5 − 660602880143

x3

F15 =x16

30+

392195

x14− 56448715

x12 +313600

143x10− 5734400

143x8 +

61931520143

x6− 330301440143

x4 +660602880

143x2

13

A∗15 =x16

30− 98

65x14 − 8232

143x12 − 219520

143x10 − 3763200

143x8 − 36126720

143x6 − 144506880

143x4

B∗15 =715

x15 +137265

x13 +98784143

x11 +2195200

143x9 +

30105600143

x7 +216760320

143x5 +

578027520143

x3

C∗15 = −x16

30− 343

195x14− 49392

715x12− 274400

143x10− 5017600

143x8− 54190080

143x6− 289013760

143x4− 578027520

143x2

D∗15 = −x16

30− 112

65x14 − 9408

143x12 − 250880

143x10 − 4300800

143x8 − 41287680

143x6 − 165150720

143x4

E∗15 =815

x15 +156865

x13 +112896

143x11 +

2508800143

x9 +34406400

143x7 +

247726080143

x5 +660602880

143x3

F ∗15 =x16

30− 392

195x14− 56448

715x12− 313600

143x10− 5734400

143x8− 61931520

143x6− 330301440

143x4− 660602880

143x2

Recurrence Formulas:∫x2n+1J2

0 (x) dx =1

2n + 1

x2n

2[(x2 + 2n2)J2

0 (x) + x2J21 (x)

]+ nx2n+1J0(x)J1(x)− 2n3

∫x2n−1J2

0 (x) dx

The next formula refers to

∫xmJ2

0 (x) dx instead of∫

xmJ21 (x) dx. To find this integral both formulas have

to be used as a system. ∫x2n+1J2

1 (x) dx =

=1

2n + 1

x2n

2[(x2 − 2n(n + 1))J2

0 (x) + x2J21 (x)

]− (n + 1)x2n+1J0(x)J1(x) + 2n2(n + 1)

∫x2n−1J2

0 (x) dx

∫

x2n+1I20 (x) dx =

12n + 1

x2n

2[(x2 − 2n2)I2

0 (x)− x2I21 (x)

]+ nx2n+1I0(x)I1(x) + 2n3

∫x2n−1I2

0 (x) dx

∫

x2n+1I21 (x) dx =

=1

2n + 1

−x2n

2[(x2 + 2n(n + 1))I2

0 (x)− x2I21 (x)

]+ (n + 1)x2n+1I0(x)I1(x) + 2n2(n + 1)

∫x2n−1I2

0 (x) dx

(See the previous remark.)

14

VIII. Integrals of the type∫

x−2nZ 2ν (x) dx

See also [4], p. 41. .Concerning the case x+2nZ2

ν (x) see XI .∫J2

0 (x)x2

dx = −(

2x +1x

)J2

0 (x) + 2 J0(x) J1(x)− 2xJ21 (x)∫

I20 (x)x2

dx =(

2x− 1x

)I20 (x)− 2 I0(x) I1(x)− 2x I2

1 (x)∫J2

1 (x)x2

dx =2x

3J2

0 (x)− 23

J0(x) J1(x) +(

2x

3− 1

3x

)J2

1 (x)∫I21 (x)x2

dx =2x

3I20 (x)− 2

3I0(x) I1(x)−

(2x

3+

13x

)I21 (x)∫

J20 (x)x4

dx =1

27x3(16x4 + 6 x2 − 9) J2

0 (x) +1

27x2(−16 x2 + 6) J0(x) J1(x) +

127x

(16x2 − 2) J21 (x)∫

I20 (x)x4

dx =1

27x3(16x4 − 6 x2 − 9) I2

0 (x)− 127x2

(16x2 + 6) I0(x) I1(x)− 127x

(16x2 + 2) I21 (x)∫

J21 (x)x4

dx =−16 x2 − 6

45xJ2

0 (x) +16 x2 − 6

45x2J0(x) J1(x) +

−16 x4 + 2 x2 − 945x3

J21 (x)∫

I21 (x)x4

dx =16 x2 − 6

45xI20 (x)− 16 x2 + 6

45x2I0(x) I1(x)− 16 x4 + 2 x2 + 9

45x3I21 (x)∫

J20 (x)x6

dx =−256 x6 − 96 x4 + 90 x2 − 675

3375x5J2

0 (x) +256 x4 − 96 x2 + 270

3375x4J0(x) J1(x)+

+−256 x4 + 32 x2 − 54

3375x3J2

1 (x)∫I20 (x)x6

dx =256 x6 − 96 x4 − 90 x2 − 675

3375x5I20 (x)− 256 x4 + 96 x2 + 270

3375x4I0(x) I1(x)−

−256 x4 + 32 x2 + 543375x3

I21 (x)∫

J21 (x)x6

dx =1

4725x5

[x2(256 x4 + 96 x2 − 90) J2

0 (x) + x(−256 x4 + 96 x2 − 270) J0(x) J1(x)+

+(256x6 − 32 x4 + 54 x2 − 675) J21 (x)

]∫

I21 (x)x6

dx =1

4725x5

[x2(256x4 − 96 x2 − 90) I2

0 (x)− x(256 x4 + 96 x2 + 270) I0(x) I1(x)−

−(256 x6 + 32 x4 + 54 x2 + 675) I21 (x)

]∫

J20 (x)x8

dx =1

385875x7

[(2048x8 + 768 x6 − 720 x4 + 3150 x2 − 55125) J2

0 (x)+

+x(−2048 x6 + 768 x4 − 2160 x2 + 15750) J0(x) J1(x) + x2 (2048x6 − 256 x4 + 432 x2 − 2250) J21 (x)

]∫

I20 (x)x8

dx =1

385875x7

[(2048x8 − 768 x6 − 720 x4 − 3150 x2 − 55125) I2

0 (x)−

−x(2048x6 + 768 x4 + 2160 x2 + 15750) I0(x) I1(x)− x2 (2048x6 + 256 x4 + 432 x2 + 2250) I21 (x)

]∫

J21 (x)x8

dx =1

496125x7

[x2(−2048 x6 − 768 x4 + 720 x2 − 3150) J2

0 (x)+

+x(2048x6 − 768 x4 + 2160 x2 − 15750) J0(x) J1(x) + (−2048 x8 + 256 x6 − 432 x4 + 2250 x2 − 55125) J21 (x)

]∫

I21 (x)x8

dx =1

496125x7

[x2(2048x6 − 768 x4 − 720 x2 − 3150) I2

0 (x)−

15

−x(2048x6 + 768 x4 + 2160 x2 + 15750) I0(x) I1(x)− (2048x8 + 256 x6 + 432 x4 + 2250 x2 + 55125) I21 (x)

]∫

J20 (x)x10

dx =1

281302875x9

[(−65536 x10 − 24576 x8 + 23040 x6 − 100800 x4 + 992250 x2 − 31255875) J2

0 (x)+

+x(65536 x8 − 24576 x6 + 69120 x4 − 504000 x2 + 6945750) J0(x) J1(x)+

+x2 (−65536 x8 + 8192 x6 − 13824 x4 + 72000 x2 − 771750) J21 (x)

]∫

I20 (x)x10

dx =1

281302875x9

[(65536x10 − 24576 x8 − 23040 x6 − 100800 x4 − 992250 x2 − 31255875) I2

0 (x)−

−x(65536 x8 + 24576x6 + 69120x4 + 504000 x2 + 6945750) I0(x) I1(x)−

−x2 (65536x8 + 8192 x6 + 13824x4 + 72000 x2 + 771750) I21 (x)

]∫

J21 (x)x10

dx =1

343814625x9

[x2(65536x8 + 24576x6 − 23040 x4 + 100800 x2 − 992250) J2

0 (x)+

+x(−65536 x8 + 24576 x6 − 69120 x4 + 504000 x2 − 6945750) J0(x) J1(x)+

+(65536x10 − 8192 x8 + 13824 x6 − 72000 x4 + 771750 x2 − 31255875) J21 (x)

]∫

I21 (x)x10

dx =1

343814625x9

[x2(65536x8 − 24576 x6 − 23040 x4 − 100800 x2 − 992250) I2

0 (x)−

−x(65536 x8 + 24576x6 + 69120x4 + 504000 x2 + 6945750) I0(x) I1(x)−

−(65536x10 + 8192 x8 + 13824 x6 + 72000 x4 + 771750 x2 − 31255875) I21 (x)

]

16

IX. Integrals of the type∫

x2nZ0(x)Z1(x) dx∫J0(x)J1(x) dx = −1

2J2

0 (x)

∫I0(x)I1(x) dx =

12I20 (x)∫

x2 · J0(x)J1(x) dx =x2

2J2

1 (x)∫x2 · I0(x)I1(x) dx =

x2

2I21 (x)∫

x4 · J0(x)J1(x) dx = −x4

6J2

0 (x) +2x3

3J0(x)J1(x) +

(x4

3− 2x2

3

)J2

1 (x)∫x4 · I0(x)I1(x) dx =

x4

6I20 (x)− 2x3

3I0(x)I1(x) +

(x4

3+

2x2

3

)I21 (x)∫

x6 · J0(x)J1(x) dx =(−x6

5+

4x4

5

)J2

0 (x) +(

6x5

5− 16x3

5

)J0(x)J1(x) +

(3x6

10− 8x4

5+

16x2

5

)J2

1 (x)∫x6 · I0(x)I1(x) dx =

(x6

5+

4x4

5

)I20 (x)−

(6x5

5+

16x3

5

)I0(x)I1(x) +

(3x6

10+

8x4

5+

16x2

5

)I21 (x)∫

x8 · J0(x)J1(x) dx =(−3x8

14+

72x6

35− 288x4

35

)J2

0 (x) +(

12x7

7− 432x5

35+

1152x3

35

)J0(x)J1(x)+

+(

2x8

7− 108x6

35+

576x4

35− 1152x2

35

)J2

1 (x)∫x8 · I0(x)I1(x) dx =

(3x8

14+

72x6

35+

288x4

35

)I20 (x)−

(12x7

7+

432x5

35+

1152x3

35

)I0(x)I1(x)+

+(

2x8

7+

108x6

35+

576x4

35+

1152x2

35

)I21 (x)∫

x10 · J0(x)J1(x) dx =(−2x10

9+

80x8

21− 256x6

7+

1024x4

7

)J2

0 (x)+

+(

20x9

9− 640x7

21+

1536x5

7− 4096x3

7

)J0(x)J1(x)+

+(

5x10

18− 320x8

63+

384x6

7− 2048x4

7+

4096x2

7

)J2

1 (x)∫x10 · I0(x)I1(x) dx =

(2x10

9+

80x8

21+

256x6

7+

1024x4

7

)I20 (x)−

−(

20x9

9+

640x7

21+

1536x5

7+

4096x3

7

)I0(x)I1(x)+

+(

5x10

18+

320x8

63+

384x6

7+

2048x4

7+

4096x2

7

)I21 (x)

Let ∫xmJ0(x)J1(x) dx = Pm(x)J2

0 (x) + Qm(x)J0(x)J1(x) + Rm(x)J21 (x)∫

xmI0(x)I1(x) dx = P ∗m(x)I20 (x) + Q∗m(x)I0(x)I1(x) + R∗m(x)I2

1 (x)

then holdsP12 = − 5

22x12 +

20033

x10 − 800077

x8 +76800

77x6 − 307200

77x4

17

Q12 =3011

x11 − 200033

x9 +64000

77x7 − 460800

77x5 +

122880077

x3

R12 =311

x12 − 25033

x10 +32000231

x8 − 11520077

x6 +614400

77x4 − 1228800

77x2

P ∗12 =522

x12 +20033

x10 +800077

x8 +76800

77x6 +

30720077

x4

Q∗ − 12 = −3011

x11 − 200033

x9 − 6400077

x7 − 46080077

x5 − 122880077

x3

R∗12 =311

x12 +25033

x10 +32000231

x8 +115200

77x6 +

61440077

x4 +1228800

77x2

P14 = − 313

x14 +1260143

x12 − 33600143

x10 +576000

143x8 − 5529600

143x6 +

22118400143

x4

Q14 =4213

x13 − 15120143

x11 +336000

143x9 − 4608000

143x7 +

33177600143

x5 − 88473600143

x3

R14 =726

x14 − 1512143

x12 +42000143

x10 − 768000143

x8 +8294400

143x6 − 44236800

143x4 +

88473600143

x2

P ∗14 =313

x14 +1260143

x12 +33600143

x10 +576000

143x8 +

5529600143

x6 +22118400

143x4

Q∗14 = −4213

x13 − 15120143

x11 − 336000143

x9 − 4608000143

x7 − 33177600143

x5 − 88473600143

x3

R∗14 =726

x14 +1512143

x12 +42000143

x10 +768000

143x8 +

8294400143

x6 +44236800

143x4 +

88473600143

x2

P16 = − 730

x16 +78465

x14 − 65856143

x12 +1756160

143x10 − 30105600

143x8 +

289013760143

x6 − 1156055040143

x4

Q16 =5615

x15 − 1097665

x13 +790272

143x11 − 17561600

143x9 +

240844800143

x7 − 1734082560143

x5 +4624220160

143x3

R16 =415

x16−2744195

x14+395136

715x12−2195200

143x10+

40140800143

x8−433520640143

x6+2312110080

143x4−4624220160

143x2

P ∗16 =730

x16 +78465

x14 +65856143

x12 +1756160

143x10 +

30105600143

x8 +289013760

143x6 +

1156055040143

x4

Q∗16 = −5615

x15− 1097665

x13− 790272143

x11− 17561600143

x9− 240844800143

x7− 1734082560143

x5− 4624220160143

x3

R∗16 =

415

x16+2744195

x14+395136

715x12+

2195200143

x10+40140800

143x8+

433520640143

x6+2312110080

143x4+

4624220160143

x2

Recurrence Formula:

(See the remark on page 14.)∫x2nJ0(x)J1(x) dx = −x2n

2J2

0 (x) + n

∫x2n−1J2

0 (x) dx

∫x2nI0(x)I1(x) dx =

x2n

2I20 (x)− n

∫x2n−1I2

0 (x) dx

18

X. Integrals of the type∫

x−(2n+1)Z0(x)Z1(x) dx

See also [4], p. 41. ∫J0(x) J1(x) dx

x= x[J 2

0 (x) + J 21 (x)]− J0(x) J1(x)∫

I0(x) I1(x) dx

x= x[I 2

0 (x)− I 21 (x)]− I0(x) I1(x)∫

J0(x) J1(x) dx

x3=

19x2

[x(−8 x2 − 3) J 2

0 (x) + (8 x2 − 3) J0(x) J1(x) + x(−8 x2 + 1) J 21 (x)

]∫

I0(x) I1(x) dx

x3=

19x2

[x(8 x2 − 3) I 2

0 (x)− (8 x2 + 3) I0(x) J1(x)− x(8 x2 + 1) I 21 (x)

]∫

J0(x) J1(x) dx

x5=

1675x4

[x(128 x4 + 48 x2 − 45) J 2

0 (x) + (−128 x4 + 48 x2 − 135) J0(x) J1(x)+

+x(128 x4 − 16 x2 + 27) J 21 (x)

]∫

I0(x) I1(x) dx

x5=

1675x4

[x(128x4 − 48 x2 − 45) I 2

0 (x)− (128 x4 + 48 x2 + 135) I0(x) I1(x)−

−x(128 x4 + 16 x2 + 27) I 21 (x)

]∫

J0(x) J1(x) dx

x7=

155 125x6

[x(−1 024x6 − 384 x4 + 360 x2 − 1 575)J 2

0 (x)+

+(1 024 x6 − 384 x4 + 1 080x2 − 7 875)J0(x) J1(x) + x (−1 024x6 + 128 x4 − 216 x2 + 1 125) J 21 (x)

]∫

I0(x) I1(x) dx

x7=

155 125x6

[x(1 024 x6 − 384 x4 − 360 x2 − 1 575) I 2

0 (x)−

−(1 024 x6 + 384 x4 + 1 080x2 + 7 875) I0(x) I1(x)− x (1 024 x6 + 128 x4 + 216 x2 + 1 125) I 21 (x)

]∫

J0(x) J1(x) dx

x9=

131 255 875x8

[x (32 768 x8 + 12 288x6 − 11 520x4 + 50 400x2 − 496 125)J 2

0 (x)+

+(−32 768x8 + 12 288x6 − 34 560x4 + 252 000x2 − 3 472 875) J0(x) J1(x)+

+x (32 768 x8 − 4 096x6 + 6 912x4 − 36 000x2 + 385 875) J 21 (x)

]∫

I0(x) I1(x) dx

x9=

131 255 875x8

[x (32 768 x8 − 12 288x6 − 11 520x4 − 50 400x2 − 496 125) I 2

0 (x)−

−(32 768 x8 + 12 288x6 + 34 560x4 + 252 000x2 + 3 472 875) I0(x) I1(x)−

−x (32 768 x8 + 4 096x6 + 6 912x4 + 36 000x2 + 385 875) I 21 (x)

]∫

J0(x) J1(x) dx

x11=

13 403 7647 875x10

[x (−1 310 720 x10−

−491 520x8 + 460 800x6 − 2 016 000 x4 + 19 845 000 x2 − 343 814 625) J 20 (x)+

+(1 310 720x10 − 491 520x8 + 1 382 400 x6 − 10 080 000 x4 + 138 915 000 x2 − 3 094 331 625) J0(x) J1(x)+

+x (−1 310 720 x10 + 163 840x8 − 276 480x6 + 1 440 000 x4 − 15 435 000 x2 + 281 302 875) J 21 (x)

]∫

I0(x) I1(x) dx

x11=

134 037 647 875x10

[x (1 310 720 x10−

−491 520x8 − 460 800x6 − 2 016 000 x4 − 19 845 000 x2 − 343 814 625) I 20 (x)−

−(1 310 720 x10 + 491 520x8 + 1 382 400 x6 + 10 080 000 x4 + 138 915 000 x2 + 3 094 331 625) I0(x) I1(x)−

−x (1 310 720 x10 + 163 840x8 + 276 480x6 + 1 440 000 x4 + 15 435 000 x2 + 281 302 875) I 21 (x)

]∫

J0(x) J1(x) dx

x13=

14 218 399 159 975x12

[x (4 194 304 x12 + 1 572 864 x10 − 1 474 560 x8+

+6451 200 x6 − 63 504 000 x4 + 1 100 206 800x2 − 29 499 294 825) J 20 (x)+

19

+(−4 194 304 x12 + 1 572 864 x10 − 4 423 680 x8 + 32 256 000 x6−

−444 528 000 x4 + 9 901 861 200x2 − 324 492 243 075) J0(x) J1(x)+

+x (4 194 304 x12 − 524 288x10 + 884 736x8 − 4 608 000 x6+

+49 392 000 x4 − 900 169 200 x2 + 24 960 941 775) J 21 (x)

]∫

I0(x) I1(x) dx

x13=

14 218 399 159 975x12

[x (4 194 304 x12 − 1 572 864 x10 − 1 474 560 x8−

−6 451 200 x6 − 63 504 000 x4 − 1 100 206 800x2 − 29 499 294 825) I 20 (x)−

−(4 194 304 x12 + 1 572 864 x10 + 4 423 680 x8 + 32 256 000 x6+

+444 528 000 x4 + 9 901 861 200x2 + 324 492 243 075) I0(x) I1(x)−

−x (4 194 304 x12 + 524 288x10 + 884 736x8 + 4 608 000 x6+

+49 392 000 x4 + 900 169 200 x2 + 24 960 941 775) I 21 (x)

]

20

XI. Integrals of the type∫

x2n J2ν (x) dx

a) The functions Θ(x) and Ω(x):

From Hankel’s asymptotic expansion of Jν(x) and Yν(x) (see [1], 9.2, or [5], XIII. A. 4) and such ofHν(x) follows, that no finite representations of the integrals

∫Z 2

ν (x) dx by functions of the type

A(x) J20 (x) + B(x) J0(x) J1(x) + C(x) J2

1 (x) + [D(x) J0(x) + E(x) J1(x)] Φ(x) + F (x) Φ 2(x)

with

A(x) =n∑

i=−m

ai x i , . . . ,

can be expected. Indeed, one has

limx→+∞

1lnx

∫ x

0

J 2ν (x) dx =

1π

and this contradicts the upper statement.At least should be given some other representations or approximations.

Θ(x) =∫ x

0

J 20 (t) dt = 2

∞∑k=0

(−1)k · (2k)!(2k + 1) · (k!)4

·(x

2

)2k+1

,

Ω(x) =∫ x

0

I 20 (t) dt = 2

∞∑k=0

(2k)!(2k + 1) · (k!)4

·(x

2

)2k+1

.

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................2

.............4

.............6

.............8

.............10

.............12

.............14

.............16

.............18

.............20x

Θ(x)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

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Figure 1 : Function Θ(x)

The dashed lines are located in the zeros of J0(x).

21

If Θ(x) is computed by its series expansion with floating point numbers with n decimal digits, then therounding error is (roughly spoken) about 10−n ·Ω(x). The computation of Ω(x) does not cause problems.

x Θ(x) Ω(x) x Θ(x) Ω(x)

1 0.850 894 480 1.186 711 080 11 1.623 448 675 27 934 437.9372 1.132 017 958 4.122 544 686 12 1.631 897 146 187 937 123.6163 1.153 502 059 16.143 998 37 13 1.653 795 366 1 274 682 776.624 1.286 956 020 77.509 947 74 14 1.696 509 451 8 704 524 383.835 1.386 983 380 425.031 292 0 15 1.706 616 878 59 786 647 515.36 1.396 339 284 2 509.864 255 16 1.719 735 792 412 698 941 831.7 1.460 064 224 15 483.965 76 17 1.755 251 443 2 861 234 688 1708 1.527 171 173 98 307.748 55 18 1.767 226 854 19 912 983 676 2449 1.534 810 723 637 083.688 6 19 1.774 861 457 139 056 981 172 080

10 1.571 266 461 4 193 041.1057 20 1.804 335 251 974 012 122 207 867

Differential equations:2xΘ′′′ ·Θ′ − 2Θ′′ ·Θ′ − xΘ′′2 + 4xΘ′2 = 0

2xΩ′′′ · Ω′ − 2Ω′′ · Ω′ − xΩ′′2 − 4xΩ′2 = 0

Asymptotic series of Θ(x) for x → +∞ :

Θ(x) ∼ 1π

[lnx +A(x) cos 2x + B(x) sin 2x + C(x)]

with

A(x) = − 12x

+29

64x3− 6747

4096x5+

1796265131072x7

− 344786683516777216x9

+2611501938675536870912x11

− 573962726457697534359738368x13

+

+8634220069330080225

1099511627776x15− 136326392392790108383875

281474976710656x17+

3417525716134419776210073759007199254740992x19

− . . . ,

B(x) = − 38x2

+195

256x4− 71505

16384x6+

26103735524288x8

− 6376138114567108864x10

+586718920037252147483648x12

− 151798966421827725137438953472x14

+

+262762002151603329375

4398046511104x16− 4692430263630584633783625

1125899906842624x18+

1312688010142958160034886062536028797018963968x20

− . . . ,

C(x) = 2.656 657 206 581 368 789 +1

16x2− 27

512x4+

3752048x6

− 385875262144x8

+11252115524288x10

− 832031392516777216x12

−

+111916712407567108864x14

− 2644032330627187534359738368x16

+1603719856835971875

34359738368x18− . . . .

A(x) =∞∑

k=1

ak

xk, B(x) =

∞∑k=1

bk

xk, C(x) =

∞∑k=0

ck

xk

k ak k bk ck

- - 0 - 2.656 657 206 581 3691 -0.500 000 000 000 000 2 -0.375 000 000 000 000 0.062 500 000 000 0003 0.453 125 000 000 000 4 0.761 718 750 000 000 -0.052 734 375 000 0005 -1.647 216 796 875 000 6 -4.364 318 847 656 250 0.183 105 468 750 0007 13.704 414 367 675 78 8 49.788 923 263 549 80 -1.471 996 307 373 0479 -205.508 877 933 025 4 10 -950.118 618 384 003 6 21.461 706 161 499 0211 4 864.301 418 280 229 12 27 321.228 759 235 24 -495.929 355 919 361 113 -167 045.138 793 094 5 14 -1 104 482.845 562 072 16 676.889 718 696 4815 7 852 777.406 997 193 16 59 745 162.196 032 50 -769 514.686 726 961 417 -484 328 639.035 390 4 18 -4 167 715 296.104 454 46 674 390.813 451 3719 37 942 157 373.010 10 20 364 344 113 252.523 3 -

A, B and C are alternating series. For some concrete x 6= 0 the item ak/xk in A should be used while|ak−2| > |ak|/x2 (see [6], 0.330). From this, one has |x| >

√|ak/ak+2| = αk as a lower boundary.

The same way bk/xk from B can be included into its sum if |x| > βk =√|bk/bk+2| and the same holds for

22

ck/xk in the case |x| > γk.

k 3 5 7 9 11 13 15 17 19αk 0.9520 1.9066 2.8844 3.8724 4.8651 5.8601 6.8564 7.8534 8.8510

k 4 6 8 10 12 14 16 18 20βk 1.4252 2.3937 3.3776 4.3684 5.3624 6.3581 7.3548 8.3521 9.3499γk 0.9186 1.8634 2.8353 3.8184 4.8070 5.7989 6.7928 7.7881 -

For example, to compute A(4.203) one gets the best value using five terms of this series because of3.8724 < 4.203 < 4.8651:

A(4.203) = −0.118 963 + 0.006 103− 0.001 256 + 0.000 591− 0.000 502 [ + 0.000 673− 0.001 308 + . . .

Let xk denote the k-th positive zero of J0(x), then holds

Θ(xk) ∼ 1π

[lnxk +

524 · x2

k

− 33129 · x4

k

+7 987

211 · x6k

− 753 375214 · x8

k

+246 293 295

218 · x10k

− . . .

]=

=1π

[lnxk +

0.312 500x2

k

− 0.646 484x4

k

+3.899 902

x6k

− 45.982 36x8

k

+939.5344

x10k

− . . .

].

Simple approximation: Θ(x) ≈ ln xπ + 0.84564:

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................x

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...............................................0.050

0.025

-0.025

-0.050

-0.075

-0.100

......... ......... ......... ......... ......... ......... .........1 3 5 7 9 11 13 15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

lnxπ + 0.84564−Θ(x)

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Figure 2

Let Mk be the k-th maximum of the difference lnx/π+0.84564−Θ(x) in xk > 1 and mk the k-th minimum.One has Mk > −mk+1 > Mk+1.

k 1 2 3 4 5 6 7 9 10 11 12 13 14xk 3.185 6.304 9.439 12.58 15.72 18.86 22.00 25.14 28.28 31.42 34.56 37.70 40.84

104 Mk 454 243 165 125 100 83.7 71.8 62.9 56.0 50.4 45.9 42.0 38.8xk · Mk 0.146 0.153 0.156 0.157 0.157 0.158 0.158 0.158 0.158 0.158 0.159 0.158 0.158

Let∆1(x) =

1π

[lnx + 2.656 . . .− cos 2x

2x− 3 sin 2x

8x2+

116x2

]−Θ(x)

∆2(x) =1π

[lnx + 2.656 . . . +

(− 1

2x+

2964x3

)cos 2x +

(− 3

8x2+

195256x4

)sin 2x +

116x2

− 27512x4

]−Θ(x)

∆3(x) =1π

[lnx + 2.656 . . . +

(− 1

2x+

2964x3

− 67474096x5

)cos 2x +

(− 3

8x2+

195256x4

− 7150516384x6

)sin 2x+

+1

16x2− 27

512x4+

3752048x6

]−Θ(x)

∆4(x) =1π

[lnx + 2.656 . . . +

(− 1

2x+

2964x3

− 67474096x5

+1796265131072x7

)cos 2x +

23

+(− 3

8x2+

195256x4

− 7150516384x6

+26103735524288x8

)sin 2x +

116x2

− 27512x4

+375

2048x6− 385875

262144x8

]−Θ(x)

Solid line: ∆1(x), Long dashes: ∆2(x), Short dashes: ∆3(x), Dots: ∆4(x):

∆1(x)

∆2(x)

∆3(x)

∆4(x)

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x

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0.005

0.010

0.015

0.020

0.025

0.030

-0.005

-0.010

-0.015

-0.020

-0.025

-0.030

................3

................5

................71

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..

..

..

..

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. . . . . . . . . . . . . . . . .

Figure 3 : Differences ∆1...4 (x), 1 ≤ x ≤ 7

24

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x

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5

10

15

20

-5

-10

-15

-20

×10−5

................

.....6.....................8

................

.....104

∆1(x)

∆2(x)

∆3(x)

∆4(x)

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.................................... . . . . . .. . .

. . .. . .

. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4 : Differences ∆1...4 (x), 4 ≤ x ≤ 10

Asymptotic behaviour of Ω(x) for x →∞:Ω(x) ∼

∼ e2x

4πx

[1 +

34x

+29

32x2+

195128x3

+6 747

2 048x4+

71 5058 192x5

+1 796 26565 536x6

+26 103 735262 144x7

+430 983 3541 048 576x8

+ . . .

]=

e2x

4πx

[1 +

0.75x

+0.90625

x2+

1.5234x3

+3.2944

x4+

8.7286x5

+27.409

x6+

99.578x7

+411.02

x8+ . . .

]

25

b) Table of Θ(x):

x 0 1 2 3 4 5 6 7 8 90.0 0.000000 0.010000 0.019999 0.029996 0.039989 0.049979 0.059964 0.069943 0.079915 0.0898790.1 0.099834 0.109778 0.119712 0.129635 0.139544 0.149439 0.159319 0.169184 0.179032 0.1888610.2 0.198673 0.208464 0.218235 0.227984 0.237711 0.247414 0.257093 0.266746 0.276373 0.2859730.3 0.295545 0.305088 0.314601 0.324083 0.333534 0.342952 0.352336 0.361687 0.371002 0.3802810.4 0.389523 0.398728 0.407894 0.417021 0.426108 0.435154 0.444158 0.453120 0.462039 0.4709130.5 0.479743 0.488527 0.497266 0.505957 0.514601 0.523196 0.531742 0.540239 0.548685 0.5570800.6 0.565424 0.573715 0.581953 0.590138 0.598269 0.606345 0.614366 0.622331 0.630239 0.6380910.7 0.645885 0.653621 0.661298 0.668917 0.676476 0.683975 0.691414 0.698792 0.706109 0.7133640.8 0.720558 0.727688 0.734756 0.741761 0.748702 0.755580 0.762393 0.769142 0.775826 0.7824450.9 0.788999 0.795487 0.801909 0.808265 0.814555 0.820779 0.826936 0.833026 0.839049 0.8450061.0 0.850894 0.856716 0.862470 0.868157 0.873776 0.879327 0.884811 0.890226 0.895575 0.9008551.1 0.906067 0.911212 0.916289 0.921298 0.926240 0.931114 0.935920 0.940659 0.945331 0.9499361.2 0.954474 0.958944 0.963348 0.967686 0.971957 0.976162 0.980301 0.984374 0.988382 0.9923241.3 0.996202 1.000015 1.003763 1.007447 1.011067 1.014624 1.018118 1.021548 1.024916 1.0282221.4 1.031466 1.034649 1.037770 1.040831 1.043832 1.046772 1.049654 1.052476 1.055240 1.0579461.5 1.060595 1.063186 1.065720 1.068199 1.070622 1.072989 1.075302 1.077561 1.079767 1.0819191.6 1.084019 1.086067 1.088064 1.090010 1.091905 1.093752 1.095549 1.097297 1.098998 1.1006521.7 1.102259 1.103820 1.105336 1.106806 1.108233 1.109617 1.110957 1.112256 1.113512 1.1147281.8 1.115904 1.117041 1.118138 1.119197 1.120219 1.121203 1.122152 1.123065 1.123943 1.1247881.9 1.125598 1.126376 1.127122 1.127837 1.128520 1.129174 1.129799 1.130395 1.130963 1.1315042.0 1.132018 1.132506 1.132970 1.133409 1.133824 1.134216 1.134586 1.134934 1.135261 1.1355682.1 1.135855 1.136123 1.136373 1.136605 1.136821 1.137020 1.137204 1.137373 1.137527 1.1376682.2 1.137796 1.137912 1.138016 1.138109 1.138192 1.138265 1.138329 1.138385 1.138433 1.1384732.3 1.138507 1.138535 1.138558 1.138575 1.138589 1.138599 1.138606 1.138610 1.138612 1.1386132.4 1.138614 1.138614 1.138614 1.138615 1.138618 1.138622 1.138629 1.138638 1.138651 1.1386682.5 1.138689 1.138715 1.138746 1.138783 1.138826 1.138876 1.138932 1.138997 1.139069 1.1391492.6 1.139239 1.139337 1.139445 1.139562 1.139690 1.139829 1.139978 1.140139 1.140312 1.1404962.7 1.140693 1.140902 1.141124 1.141360 1.141608 1.141871 1.142147 1.142438 1.142743 1.1430632.8 1.143398 1.143748 1.144113 1.144494 1.144891 1.145303 1.145732 1.146177 1.146639 1.1471172.9 1.147612 1.148123 1.148652 1.149197 1.149760 1.150340 1.150938 1.151552 1.152185 1.1528353.0 1.153502 1.154187 1.154890 1.155610 1.156349 1.157104 1.157878 1.158669 1.159478 1.1603053.1 1.161149 1.162011 1.162890 1.163787 1.164701 1.165633 1.166581 1.167547 1.168530 1.1695303.2 1.170547 1.171581 1.172631 1.173697 1.174780 1.175880 1.176995 1.178126 1.179273 1.1804353.3 1.181613 1.182806 1.184014 1.185237 1.186474 1.187726 1.188992 1.190272 1.191566 1.1928733.4 1.194193 1.195527 1.196873 1.198232 1.199604 1.200987 1.202382 1.203789 1.205207 1.2066363.5 1.208076 1.209526 1.210986 1.212456 1.213936 1.215425 1.216923 1.218430 1.219945 1.2214683.6 1.222999 1.224538 1.226084 1.227636 1.229195 1.230760 1.232332 1.233908 1.235491 1.2370783.7 1.238669 1.240265 1.241865 1.243469 1.245076 1.246685 1.248298 1.249913 1.251530 1.2531483.8 1.254768 1.256389 1.258011 1.259633 1.261255 1.262877 1.264498 1.266119 1.267738 1.2693553.9 1.270971 1.272584 1.274195 1.275804 1.277409 1.279010 1.280608 1.282202 1.283791 1.2853764.0 1.286956 1.288531 1.290100 1.291663 1.293220 1.294771 1.296315 1.297852 1.299382 1.3009044.1 1.302419 1.303925 1.305424 1.306913 1.308394 1.309866 1.311329 1.312782 1.314225 1.3156594.2 1.317082 1.318495 1.319897 1.321288 1.322668 1.324037 1.325394 1.326740 1.328073 1.3293954.3 1.330705 1.332002 1.333286 1.334558 1.335817 1.337062 1.338295 1.339514 1.340720 1.3419124.4 1.343090 1.344255 1.345405 1.346541 1.347664 1.348771 1.349865 1.350944 1.352008 1.3530574.5 1.354092 1.355112 1.356118 1.357108 1.358083 1.359043 1.359989 1.360919 1.361833 1.3627334.6 1.363618 1.364487 1.365341 1.366180 1.367004 1.367813 1.368606 1.369384 1.370147 1.3708954.7 1.371628 1.372346 1.373049 1.373737 1.374410 1.375068 1.375712 1.376341 1.376955 1.3775554.8 1.378140 1.378711 1.379267 1.379810 1.380338 1.380853 1.381353 1.381840 1.382314 1.3827734.9 1.383220 1.383653 1.384073 1.384481 1.384875 1.385257 1.385627 1.385984 1.386329 1.3866625.0 1.386983 1.387293 1.387591 1.387878 1.388154 1.388419 1.388673 1.388916 1.389150 1.3893735.1 1.389586 1.389790 1.389984 1.390168 1.390344 1.390510 1.390668 1.390818 1.390959 1.3910935.2 1.391218 1.391336 1.391446 1.391550 1.391646 1.391736 1.391820 1.391897 1.391968 1.3920345.3 1.392094 1.392149 1.392199 1.392244 1.392285 1.392321 1.392353 1.392381 1.392406 1.3924285.4 1.392446 1.392462 1.392475 1.392486 1.392494 1.392501 1.392506 1.392509 1.392512 1.3925135.5 1.392514 1.392514 1.392514 1.392514 1.392514 1.392515 1.392516 1.392519 1.392522 1.3925275.6 1.392533 1.392542 1.392552 1.392564 1.392579 1.392597 1.392617 1.392641 1.392668 1.3926985.7 1.392732 1.392770 1.392812 1.392858 1.392909 1.392964 1.393024 1.393089 1.393160 1.3932365.8 1.393317 1.393404 1.393497 1.393596 1.393701 1.393812 1.393930 1.394054 1.394186 1.3943245.9 1.394469 1.394622 1.394782 1.394949 1.395124 1.395307 1.395497 1.395696 1.395902 1.396116

26

Θ(x):

x 0 1 2 3 4 5 6 7 8 96.0 1.396339 1.396570 1.396810 1.397058 1.397315 1.397580 1.397854 1.398137 1.398429 1.3987306.1 1.399039 1.399358 1.399686 1.400023 1.400370 1.400725 1.401090 1.401465 1.401848 1.4022416.2 1.402643 1.403055 1.403476 1.403907 1.404347 1.404796 1.405255 1.405723 1.406201 1.4066886.3 1.407184 1.407689 1.408204 1.408728 1.409262 1.409804 1.410356 1.410916 1.411486 1.4120656.4 1.412652 1.413249 1.413854 1.414468 1.415090 1.415721 1.416361 1.417009 1.417665 1.4183296.5 1.419002 1.419682 1.420371 1.421067 1.421771 1.422482 1.423201 1.423927 1.424661 1.4254016.6 1.426149 1.426903 1.427664 1.428432 1.429206 1.429987 1.430773 1.431566 1.432365 1.4331696.7 1.433979 1.434794 1.435615 1.436440 1.437271 1.438107 1.438947 1.439791 1.440640 1.4414936.8 1.442351 1.443212 1.444076 1.444944 1.445816 1.446690 1.447568 1.448448 1.449331 1.4502166.9 1.451104 1.451993 1.452885 1.453778 1.454673 1.455569 1.456466 1.457365 1.458264 1.4591647.0 1.460064 1.460965 1.461865 1.462766 1.463666 1.464566 1.465466 1.466364 1.467262 1.4681587.1 1.469053 1.469947 1.470838 1.471728 1.472616 1.473502 1.474386 1.475267 1.476145 1.4770207.2 1.477892 1.478761 1.479627 1.480489 1.481347 1.482202 1.483053 1.483899 1.484741 1.4855797.3 1.486412 1.487240 1.488064 1.488882 1.489695 1.490503 1.491305 1.492102 1.492893 1.4936787.4 1.494457 1.495231 1.495997 1.496758 1.497512 1.498259 1.499000 1.499734 1.500461 1.5011817.5 1.501894 1.502600 1.503298 1.503989 1.504673 1.505349 1.506017 1.506678 1.507331 1.5079767.6 1.508613 1.509242 1.509863 1.510476 1.511080 1.511677 1.512265 1.512844 1.513416 1.5139797.7 1.514533 1.515079 1.515616 1.516145 1.516665 1.517177 1.517680 1.518174 1.518660 1.5191377.8 1.519605 1.520065 1.520516 1.520958 1.521392 1.521817 1.522234 1.522641 1.523040 1.5234317.9 1.523813 1.524187 1.524552 1.524908 1.525256 1.525596 1.525928 1.526251 1.526566 1.5268728.0 1.527171 1.527462 1.527744 1.528019 1.528286 1.528545 1.528796 1.529040 1.529276 1.5295058.1 1.529726 1.529940 1.530147 1.530347 1.530539 1.530725 1.530904 1.531076 1.531241 1.5314008.2 1.531553 1.531699 1.531839 1.531973 1.532101 1.532223 1.532340 1.532450 1.532556 1.5326568.3 1.532750 1.532840 1.532925 1.533005 1.533080 1.533150 1.533216 1.533278 1.533336 1.5333898.4 1.533439 1.533485 1.533527 1.533566 1.533602 1.533634 1.533664 1.533690 1.533714 1.5337358.5 1.533754 1.533771 1.533785 1.533797 1.533808 1.533816 1.533824 1.533830 1.533834 1.5338388.6 1.533840 1.533842 1.533843 1.533844 1.533844 1.533844 1.533844 1.533844 1.533844 1.5338458.7 1.533846 1.533848 1.533851 1.533855 1.533860 1.533866 1.533873 1.533882 1.533893 1.5339058.8 1.533920 1.533936 1.533955 1.533976 1.533999 1.534025 1.534054 1.534086 1.534120 1.5341588.9 1.534199 1.534243 1.534291 1.534342 1.534397 1.534456 1.534519 1.534585 1.534656 1.5347319.0 1.534811 1.534895 1.534983 1.535076 1.535174 1.535276 1.535384 1.535496 1.535613 1.5357369.1 1.535864 1.535997 1.536136 1.536280 1.536429 1.536584 1.536745 1.536912 1.537084 1.5372629.2 1.537446 1.537636 1.537832 1.538034 1.538242 1.538456 1.538677 1.538903 1.539136 1.5393759.3 1.539621 1.539873 1.540131 1.540395 1.540666 1.540943 1.541227 1.541517 1.541813 1.5421169.4 1.542425 1.542741 1.543063 1.543391 1.543726 1.544068 1.544415 1.544769 1.545130 1.5454969.5 1.545869 1.546249 1.546634 1.547026 1.547423 1.547827 1.548237 1.548653 1.549075 1.5495039.6 1.549937 1.550377 1.550822 1.551273 1.551730 1.552192 1.552660 1.553133 1.553612 1.5540969.7 1.554586 1.555080 1.555580 1.556084 1.556594 1.557108 1.557628 1.558151 1.558680 1.5592139.8 1.559750 1.560292 1.560838 1.561388 1.561942 1.562500 1.563061 1.563627 1.564196 1.5647689.9 1.565344 1.565924 1.566506 1.567092 1.567680 1.568271 1.568865 1.569462 1.570061 1.57066310.0 1.571266 1.571872 1.572480 1.573090 1.573702 1.574315 1.574930 1.575546 1.576163 1.57678210.1 1.577402 1.578022 1.578644 1.579266 1.579888 1.580511 1.581135 1.581758 1.582382 1.58300510.2 1.583628 1.584251 1.584874 1.585495 1.586117 1.586737 1.587356 1.587975 1.588592 1.58920810.3 1.589822 1.590435 1.591046 1.591655 1.592263 1.592868 1.593472 1.594073 1.594671 1.59526810.4 1.595861 1.596452 1.597040 1.597626 1.598208 1.598787 1.599363 1.599935 1.600504 1.60107010.5 1.601632 1.602190 1.602744 1.603294 1.603841 1.604383 1.604921 1.605455 1.605984 1.60650910.6 1.607030 1.607546 1.608057 1.608563 1.609065 1.609562 1.610053 1.610540 1.611021 1.61149810.7 1.611969 1.612435 1.612895 1.613350 1.613800 1.614244 1.614682 1.615115 1.615542 1.61596410.8 1.616380 1.616790 1.617194 1.617592 1.617985 1.618371 1.618752 1.619127 1.619495 1.61985810.9 1.620215 1.620565 1.620910 1.621249 1.621581 1.621908 1.622228 1.622542 1.622850 1.62315311.0 1.623449 1.623739 1.624023 1.624301 1.624573 1.624839 1.625099 1.625353 1.625601 1.62584311.1 1.626079 1.626310 1.626534 1.626753 1.626966 1.627174 1.627375 1.627572 1.627762 1.62794711.2 1.628127 1.628301 1.628470 1.628633 1.628792 1.628945 1.629092 1.629235 1.629373 1.62950611.3 1.629634 1.629757 1.629876 1.629990 1.630099 1.630203 1.630304 1.630400 1.630491 1.63057911.4 1.630662 1.630742 1.630817 1.630889 1.630957 1.631021 1.631082 1.631139 1.631193 1.63124311.5 1.631290 1.631335 1.631376 1.631414 1.631450 1.631483 1.631513 1.631541 1.631566 1.63158911.6 1.631610 1.631629 1.631646 1.631661 1.631674 1.631686 1.631696 1.631705 1.631712 1.63171911.7 1.631724 1.631728 1.631731 1.631733 1.631735 1.631736 1.631737 1.631737 1.631737 1.63173711.8 1.631738 1.631738 1.631738 1.631739 1.631740 1.631741 1.631743 1.631746 1.631750 1.63175511.9 1.631760 1.631767 1.631775 1.631785 1.631796 1.631808 1.631822 1.631838 1.631856 1.631875

27

Θ(x):

x 0 1 2 3 4 5 6 7 8 912.0 1.631897 1.631921 1.631947 1.631975 1.632006 1.632039 1.632075 1.632114 1.632155 1.63219912.1 1.632246 1.632296 1.632349 1.632406 1.632465 1.632528 1.632594 1.632664 1.632737 1.63281412.2 1.632895 1.632979 1.633067 1.633159 1.633255 1.633355 1.633459 1.633567 1.633679 1.63379512.3 1.633916 1.634041 1.634170 1.634304 1.634442 1.634584 1.634731 1.634883 1.635039 1.63520012.4 1.635366 1.635536 1.635711 1.635891 1.636075 1.636264 1.636458 1.636657 1.636861 1.63706912.5 1.637282 1.637501 1.637724 1.637951 1.638184 1.638422 1.638665 1.638912 1.639164 1.63942112.6 1.639683 1.639950 1.640222 1.640498 1.640780 1.641066 1.641356 1.641652 1.641952 1.64225712.7 1.642566 1.642881 1.643199 1.643523 1.643850 1.644183 1.644519 1.644860 1.645206 1.64555512.8 1.645909 1.646268 1.646630 1.646996 1.647367 1.647741 1.648120 1.648502 1.648888 1.64927812.9 1.649672 1.650069 1.650469 1.650874 1.651281 1.651692 1.652107 1.652524 1.652945 1.65336913.0 1.653795 1.654225 1.654657 1.655093 1.655531 1.655971 1.656414 1.656859 1.657307 1.65775713.1 1.658209 1.658664 1.659120 1.659578 1.660038 1.660500 1.660963 1.661428 1.661894 1.66236213.2 1.662831 1.663301 1.663772 1.664244 1.664717 1.665191 1.665665 1.666141 1.666616 1.66709213.3 1.667569 1.668045 1.668522 1.668999 1.669476 1.669952 1.670429 1.670905 1.671380 1.67185513.4 1.672330 1.672803 1.673276 1.673748 1.674219 1.674689 1.675158 1.675625 1.676091 1.67655513.5 1.677018 1.677480 1.677939 1.678397 1.678853 1.679307 1.679759 1.680209 1.680656 1.68110213.6 1.681544 1.681985 1.682422 1.682857 1.683290 1.683719 1.684146 1.684570 1.684991 1.68540913.7 1.685823 1.686235 1.686643 1.687047 1.687449 1.687847 1.688241 1.688632 1.689019 1.68940213.8 1.689782 1.690157 1.690529 1.690897 1.691261 1.691621 1.691977 1.692329 1.692676 1.69302013.9 1.693359 1.693694 1.694024 1.694350 1.694672 1.694990 1.695303 1.695611 1.695915 1.69621514.0 1.696509 1.696800 1.697086 1.697367 1.697643 1.697915 1.698183 1.698445 1.698703 1.69895714.1 1.699205 1.699449 1.699689 1.699923 1.700153 1.700379 1.700599 1.700815 1.701027 1.70123414.2 1.701436 1.701633 1.701826 1.702015 1.702199 1.702378 1.702553 1.702723 1.702889 1.70305114.3 1.703208 1.703361 1.703509 1.703653 1.703793 1.703929 1.704061 1.704188 1.704311 1.70443114.4 1.704546 1.704658 1.704765 1.704869 1.704969 1.705065 1.705157 1.705246 1.705332 1.70541314.5 1.705492 1.705567 1.705638 1.705707 1.705772 1.705834 1.705893 1.705949 1.706002 1.70605214.6 1.706100 1.706144 1.706186 1.706226 1.706263 1.706297 1.706330 1.706360 1.706388 1.70641314.7 1.706437 1.706459 1.706479 1.706497 1.706513 1.706528 1.706541 1.706553 1.706563 1.70657214.8 1.706580 1.706587 1.706593 1.706598 1.706601 1.706605 1.706607 1.706609 1.706610 1.70661114.9 1.706612 1.706612 1.706612 1.706612 1.706612 1.706612 1.706613 1.706613 1.706614 1.70661515.0 1.706617 1.706619 1.706622 1.706626 1.706631 1.706636 1.706642 1.706650 1.706659 1.70666915.1 1.706680 1.706693 1.706707 1.706722 1.706740 1.706759 1.706779 1.706802 1.706827 1.70685315.2 1.706882 1.706912 1.706945 1.706980 1.707018 1.707058 1.707100 1.707145 1.707192 1.70724315.3 1.707295 1.707351 1.707409 1.707471 1.707535 1.707602 1.707672 1.707746 1.707822 1.70790215.4 1.707985 1.708071 1.708160 1.708253 1.708349 1.708449 1.708552 1.708659 1.708769 1.70888315.5 1.709001 1.709122 1.709247 1.709375 1.709507 1.709643 1.709783 1.709927 1.710074 1.71022615.6 1.710381 1.710540 1.710703 1.710869 1.711040 1.711215 1.711393 1.711576 1.711762 1.71195215.7 1.712147 1.712345 1.712547 1.712753 1.712963 1.713177 1.713394 1.713616 1.713841 1.71407115.8 1.714304 1.714541 1.714782 1.715026 1.715275 1.715527 1.715782 1.716042 1.716305 1.71657215.9 1.716842 1.717116 1.717394 1.717675 1.717959 1.718247 1.718538 1.718833 1.719130 1.71943116.0 1.719736 1.720043 1.720354 1.720667 1.720984 1.721304 1.721626 1.721952 1.722280 1.72261116.1 1.722945 1.723281 1.723620 1.723961 1.724305 1.724651 1.725000 1.725351 1.725704 1.72605916.2 1.726416 1.726775 1.727136 1.727499 1.727864 1.728231 1.728599 1.728969 1.729340 1.72971316.3 1.730087 1.730463 1.730839 1.731217 1.731596 1.731976 1.732357 1.732738 1.733121 1.73350416.4 1.733888 1.734272 1.734657 1.735042 1.735428 1.735813 1.736199 1.736585 1.736971 1.73735716.5 1.737743 1.738128 1.738514 1.738898 1.739283 1.739667 1.740050 1.740433 1.740814 1.74119516.6 1.741575 1.741954 1.742332 1.742709 1.743085 1.743459 1.743832 1.744204 1.744574 1.74494316.7 1.745310 1.745675 1.746039 1.746400 1.746760 1.747118 1.747474 1.747827 1.748179 1.74852816.8 1.748875 1.749220 1.749562 1.749902 1.750239 1.750574 1.750906 1.751235 1.751562 1.75188616.9 1.752207 1.752525 1.752841 1.753153 1.753462 1.753768 1.754071 1.754371 1.754668 1.75496117.0 1.755251 1.755538 1.755822 1.756102 1.756379 1.756652 1.756922 1.757188 1.757450 1.75771017.1 1.757965 1.758217 1.758465 1.758710 1.758951 1.759188 1.759422 1.759651 1.759877 1.76010017.2 1.760318 1.760533 1.760744 1.760951 1.761155 1.761354 1.761550 1.761742 1.761930 1.76211517.3 1.762295 1.762472 1.762645 1.762815 1.762980 1.763142 1.763300 1.763455 1.763605 1.76375217.4 1.763896 1.764035 1.764171 1.764304 1.764433 1.764558 1.764680 1.764798 1.764913 1.76502417.5 1.765132 1.765237 1.765338 1.765436 1.765531 1.765623 1.765711 1.765796 1.765878 1.76595717.6 1.766033 1.766106 1.766176 1.766244 1.766308 1.766370 1.766429 1.766485 1.766538 1.76659017.7 1.766638 1.766684 1.766728 1.766769 1.766808 1.766845 1.766880 1.766912 1.766943 1.76697117.8 1.766998 1.767023 1.767046 1.767067 1.767086 1.767104 1.767121 1.767136 1.767149 1.76716117.9 1.767172 1.767182 1.767191 1.767198 1.767205 1.767210 1.767215 1.767219 1.767222 1.767225

28

Θ(x):

x 0 1 2 3 4 5 6 7 8 918.0 1.767227 1.767228 1.767230 1.767230 1.767231 1.767231 1.767231 1.767231 1.767231 1.76723118.1 1.767231 1.767232 1.767232 1.767233 1.767235 1.767237 1.767239 1.767242 1.767246 1.76725118.2 1.767256 1.767262 1.767269 1.767278 1.767287 1.767297 1.767309 1.767322 1.767336 1.76735218.3 1.767369 1.767388 1.767408 1.767430 1.767454 1.767479 1.767506 1.767536 1.767567 1.76760018.4 1.767635 1.767672 1.767711 1.767753 1.767797 1.767843 1.767891 1.767942 1.767995 1.76805118.5 1.768109 1.768170 1.768233 1.768299 1.768368 1.768440 1.768514 1.768591 1.768671 1.76875418.6 1.768840 1.768928 1.769020 1.769115 1.769212 1.769313 1.769417 1.769523 1.769633 1.76974718.7 1.769863 1.769982 1.770105 1.770231 1.770360 1.770492 1.770628 1.770766 1.770909 1.77105418.8 1.771203 1.771354 1.771510 1.771668 1.771830 1.771995 1.772163 1.772335 1.772510 1.77268818.9 1.772870 1.773054 1.773242 1.773434 1.773628 1.773826 1.774026 1.774230 1.774438 1.77464819.0 1.774861 1.775078 1.775298 1.775520 1.775746 1.775975 1.776206 1.776441 1.776679 1.77691919.1 1.777162 1.777408 1.777657 1.777909 1.778163 1.778420 1.778680 1.778942 1.779207 1.77947419.2 1.779744 1.780016 1.780290 1.780567 1.780846 1.781128 1.781411 1.781697 1.781985 1.78227519.3 1.782566 1.782860 1.783156 1.783453 1.783752 1.784053 1.784356 1.784660 1.784966 1.78527319.4 1.785582 1.785892 1.786203 1.786515 1.786829 1.787144 1.787460 1.787777 1.788094 1.78841319.5 1.788733 1.789053 1.789374 1.789695 1.790017 1.790340 1.790663 1.790986 1.791310 1.79163419.6 1.791958 1.792282 1.792607 1.792931 1.793255 1.793579 1.793903 1.794226 1.794549 1.79487219.7 1.795194 1.795516 1.795837 1.796157 1.796477 1.796796 1.797114 1.797431 1.797747 1.79806219.8 1.798376 1.798689 1.799000 1.799311 1.799619 1.799927 1.800233 1.800538 1.800841 1.80114219.9 1.801442 1.801740 1.802036 1.802330 1.802623 1.802913 1.803202 1.803488 1.803773 1.80405520.0 1.804335 1.804613 1.804889 1.805162 1.805433 1.805701 1.805967 1.806231 1.806492 1.80675020.1 1.807006 1.807259 1.807510 1.807758 1.808003 1.808245 1.808484 1.808721 1.808955 1.80918620.2 1.809414 1.809639 1.809861 1.810080 1.810296 1.810509 1.810719 1.810926 1.811129 1.81133020.3 1.811528 1.811722 1.811913 1.812101 1.812286 1.812468 1.812647 1.812822 1.812994 1.81316320.4 1.813329 1.813492 1.813651 1.813808 1.813961 1.814111 1.814257 1.814401 1.814541 1.81467820.5 1.814812 1.814943 1.815071 1.815196 1.815317 1.815436 1.815551 1.815664 1.815773 1.81587920.6 1.815983 1.816083 1.816181 1.816275 1.816367 1.816456 1.816542 1.816625 1.816706 1.81678420.7 1.816859 1.816931 1.817001 1.817068 1.817133 1.817195 1.817254 1.817312 1.817367 1.81741920.8 1.817469 1.817517 1.817563 1.817606 1.817648 1.817687 1.817724 1.817759 1.817792 1.81782420.9 1.817853 1.817881 1.817907 1.817932 1.817954 1.817975 1.817995 1.818013 1.818030 1.81804521.0 1.818059 1.818072 1.818083 1.818094 1.818103 1.818111 1.818119 1.818125 1.818131 1.81813621.1 1.818140 1.818143 1.818146 1.818148 1.818150 1.818151 1.818152 1.818153 1.818153 1.81815421.2 1.818154 1.818154 1.818154 1.818154 1.818154 1.818154 1.818155 1.818156 1.818157 1.81815921.3 1.818161 1.818163 1.818166 1.818170 1.818175 1.818180 1.818186 1.818193 1.818201 1.81821021.4 1.818220 1.818231 1.818243 1.818256 1.818271 1.818287 1.818304 1.818323 1.818343 1.81836421.5 1.818387 1.818412 1.818439 1.818467 1.818497 1.818528 1.818562 1.818597 1.818634 1.81867421.6 1.818715 1.818758 1.818804 1.818851 1.818901 1.818953 1.819007 1.819063 1.819122 1.81918321.7 1.819246 1.819312 1.819380 1.819451 1.819524 1.819600 1.819678 1.819759 1.819842 1.81992821.8 1.820017 1.820108 1.820202 1.820299 1.820398 1.820500 1.820605 1.820713 1.820823 1.82093621.9 1.821052 1.821171 1.821293 1.821417 1.821544 1.821674 1.821807 1.821943 1.822081 1.82222222.0 1.822367 1.822514 1.822663 1.822816 1.822971 1.823130 1.823291 1.823455 1.823621 1.82379122.1 1.823963 1.824138 1.824315 1.824495 1.824678 1.824864 1.825052 1.825243 1.825437 1.82563322.2 1.825832 1.826033 1.826237 1.826443 1.826652 1.826863 1.827077 1.827293 1.827511 1.82773222.3 1.827955 1.828180 1.828407 1.828637 1.828869 1.829102 1.829338 1.829576 1.829816 1.83005822.4 1.830301 1.830547 1.830794 1.831043 1.831294 1.831547 1.831801 1.832057 1.832314 1.83257322.5 1.832833 1.833095 1.833358 1.833622 1.833888 1.834154 1.834422 1.834691 1.834961 1.83523222.6 1.835504 1.835777 1.836051 1.836325 1.836600 1.836876 1.837152 1.837429 1.837707 1.83798522.7 1.838263 1.838542 1.838821 1.839100 1.839379 1.839659 1.839938 1.840218 1.840497 1.84077622.8 1.841056 1.841334 1.841613 1.841891 1.842169 1.842447 1.842724 1.843000 1.843276 1.84355122.9 1.843826 1.844099 1.844372 1.844644 1.844915 1.845185 1.845455 1.845722 1.845989 1.84625523.0 1.846519 1.846783 1.847044 1.847305 1.847564 1.847821 1.848077 1.848332 1.848584 1.84883623.1 1.849085 1.849333 1.849579 1.849823 1.850065 1.850305 1.850544 1.850780 1.851014 1.85124723.2 1.851477 1.851705 1.851931 1.852154 1.852376 1.852595 1.852812 1.853026 1.853239 1.85344823.3 1.853656 1.853861 1.854063 1.854263 1.854461 1.854656 1.854848 1.855038 1.855225 1.85541023.4 1.855592 1.855771 1.855948 1.856122 1.856293 1.856462 1.856627 1.856791 1.856951 1.85710923.5 1.857264 1.857416 1.857565 1.857712 1.857856 1.857997 1.858136 1.858271 1.858404 1.85853423.6 1.858662 1.858786 1.858908 1.859027 1.859144 1.859258 1.859369 1.859477 1.859583 1.85968623.7 1.859786 1.859884 1.859979 1.860071 1.860161 1.860249 1.860333 1.860416 1.860496 1.86057323.8 1.860648 1.860720 1.860790 1.860858 1.860923 1.860987 1.861047 1.861106 1.861162 1.86121623.9 1.861268 1.861318 1.861366 1.861412 1.861455 1.861497 1.861537 1.861575 1.861611 1.861645

29

Θ(x):

x 0 1 2 3 4 5 6 7 8 924.0 1.861678 1.861709 1.861738 1.861765 1.861791 1.861815 1.861838 1.861859 1.861879 1.86189724.1 1.861915 1.861930 1.861945 1.861958 1.861971 1.861982 1.861992 1.862001 1.862009 1.86201724.2 1.862023 1.862029 1.862034 1.862038 1.862042 1.862045 1.862047 1.862049 1.862051 1.86205224.3 1.862053 1.862053 1.862054 1.862054 1.862054 1.862054 1.862054 1.862054 1.862054 1.86205524.4 1.862055 1.862056 1.862057 1.862058 1.862060 1.862062 1.862065 1.862068 1.862072 1.86207724.5 1.862082 1.862088 1.862095 1.862102 1.862111 1.862120 1.862131 1.862142 1.862155 1.86216924.6 1.862184 1.862200 1.862217 1.862236 1.862256 1.862278 1.862301 1.862325 1.862351 1.86237824.7 1.862407 1.862438 1.862471 1.862505 1.862541 1.862578 1.862618 1.862659 1.862702 1.86274724.8 1.862795 1.862844 1.862895 1.862948 1.863003 1.863060 1.863120 1.863181 1.863245 1.86331124.9 1.863379 1.863450 1.863522 1.863597 1.863675 1.863754 1.863836 1.863920 1.864007 1.86409625.0 1.864188 1.864282 1.864378 1.864477 1.864578 1.864681 1.864788 1.864896 1.865007 1.86512125.1 1.865237 1.865355 1.865476 1.865600 1.865725 1.865854 1.865985 1.866118 1.866254 1.86639225.2 1.866533 1.866676 1.866822 1.866970 1.867120 1.867273 1.867429 1.867586 1.867746 1.86790925.3 1.868074 1.868241 1.868410 1.868582 1.868756 1.868932 1.869110 1.869291 1.869473 1.86965825.4 1.869845 1.870034 1.870226 1.870419 1.870614 1.870811 1.871010 1.871211 1.871414 1.87161925.5 1.871826 1.872034 1.872244 1.872456 1.872670 1.872885 1.873102 1.873320 1.873540 1.87376225.6 1.873985 1.874209 1.874435 1.874662 1.874890 1.875119 1.875350 1.875582 1.875815 1.87604925.7 1.876284 1.876520 1.876757 1.876995 1.877233 1.877473 1.877713 1.877954 1.878196 1.87843825.8 1.878681 1.878924 1.879167 1.879412 1.879656 1.879901 1.880146 1.880391 1.880637 1.88088225.9 1.881128 1.881373 1.881619 1.881864 1.882110 1.882355 1.882600 1.882845 1.883089 1.88333326.0 1.883577 1.883820 1.884062 1.884304 1.884546 1.884787 1.885027 1.885266 1.885504 1.88574226.1 1.885979 1.886215 1.886450 1.886683 1.886916 1.887148 1.887378 1.887608 1.887836 1.88806226.2 1.888288 1.888512 1.888735 1.888956 1.889175 1.889394 1.889610 1.889825 1.890039 1.89025126.3 1.890461 1.890669 1.890876 1.891080 1.891283 1.891484 1.891683 1.891881 1.892076 1.89226926.4 1.892460 1.892650 1.892837 1.893022 1.893205 1.893386 1.893564 1.893741 1.893915 1.89408726.5 1.894257 1.894425 1.894590 1.894753 1.894913 1.895072 1.895228 1.895381 1.895533 1.89568226.6 1.895828 1.895972 1.896114 1.896253 1.896390 1.896525 1.896657 1.896787 1.896914 1.89703926.7 1.897161 1.897281 1.897398 1.897514 1.897626 1.897736 1.897844 1.897950 1.898053 1.89815326.8 1.898252 1.898348 1.898441 1.898532 1.898621 1.898708 1.898792 1.898874 1.898953 1.89903126.9 1.899106 1.899179 1.899250 1.899318 1.899384 1.899449 1.899511 1.899571 1.899629 1.89968527.0 1.899739 1.899791 1.899841 1.899889 1.899935 1.899979 1.900021 1.900062 1.900101 1.90013827.1 1.900173 1.900207 1.900239 1.900270 1.900298 1.900326 1.900352 1.900376 1.900399 1.90042127.2 1.900441 1.900460 1.900477 1.900494 1.900509 1.900523 1.900536 1.900548 1.900559 1.90056927.3 1.900578 1.900587 1.900594 1.900600 1.900606 1.900611 1.900616 1.900620 1.900623 1.90062627.4 1.900628 1.900630 1.900631 1.900632 1.900633 1.900634 1.900634 1.900634 1.900634 1.90063427.5 1.900634 1.900634 1.900634 1.900635 1.900635 1.900636 1.900636 1.900638 1.900639 1.90064127.6 1.900643 1.900646 1.900650 1.900654 1.900658 1.900664 1.900669 1.900676 1.900684 1.90069227.7 1.900701 1.900711 1.900722 1.900735 1.900748 1.900762 1.900777 1.900794 1.900811 1.90083027.8 1.900851 1.900872 1.900895 1.900919 1.900945 1.900972 1.901001 1.901031 1.901062 1.90109627.9 1.901130 1.901167 1.901205 1.901245 1.901287 1.901330 1.901375 1.901422 1.901471 1.90152228.0 1.901574 1.901629 1.901685 1.901743 1.901804 1.901866 1.901931 1.901997 1.902065 1.90213628.1 1.902208 1.902283 1.902360 1.902439 1.902520 1.902603 1.902688 1.902776 1.902865 1.90295728.2 1.903051 1.903148 1.903246 1.903347 1.903450 1.903555 1.903662 1.903772 1.903883 1.90399728.3 1.904113 1.904232 1.904352 1.904475 1.904600 1.904727 1.904856 1.904988 1.905121 1.90525728.4 1.905395 1.905535 1.905677 1.905821 1.905968 1.906116 1.906267 1.906419 1.906574 1.90673028.5 1.906889 1.907049 1.907212 1.907376 1.907542 1.907710 1.907880 1.908052 1.908226 1.90840128.6 1.908578 1.908757 1.908937 1.909120 1.909303 1.909489 1.909676 1.909864 1.910054 1.91024628.7 1.910439 1.910633 1.910829 1.911026 1.911224 1.911424 1.911625 1.911827 1.912030 1.91223528.8 1.912440 1.912646 1.912854 1.913062 1.913272 1.913482 1.913693 1.913905 1.914118 1.91433128.9 1.914545 1.914760 1.914975 1.915191 1.915407 1.915624 1.915841 1.916059 1.916277 1.91649529.0 1.916713 1.916932 1.917151 1.917370 1.917589 1.917808 1.918027 1.918246 1.918465 1.91868429.1 1.918902 1.919121 1.919339 1.919556 1.919774 1.919991 1.920207 1.920423 1.920639 1.92085429.2 1.921068 1.921282 1.921495 1.921707 1.921918 1.922129 1.922339 1.922547 1.922755 1.92296229.3 1.923168 1.923373 1.923577 1.923779 1.923981 1.924181 1.924380 1.924578 1.924774 1.92496929.4 1.925163 1.925355 1.925546 1.925735 1.925923 1.926109 1.926294 1.926477 1.926659 1.92683829.5 1.927017 1.927193 1.927368 1.927541 1.927712 1.927881 1.928049 1.928214 1.928378 1.92854029.6 1.928700 1.928858 1.929014 1.929168 1.929320 1.929470 1.929618 1.929764 1.929908 1.93004929.7 1.930189 1.930327 1.930462 1.930596 1.930727 1.930856 1.930983 1.931108 1.931231 1.93135129.8 1.931470 1.931586 1.931700 1.931812 1.931922 1.932029 1.932135 1.932238 1.932339 1.93243829.9 1.932535 1.932629 1.932722 1.932812 1.932900 1.932986 1.933070 1.933152 1.933232 1.933309

30

Θ(x):

x 0 1 2 3 4 5 6 7 8 930.0 1.933385 1.933459 1.933530 1.933600 1.933667 1.933733 1.933796 1.933858 1.933918 1.93397530.1 1.934031 1.934085 1.934137 1.934188 1.934236 1.934283 1.934328 1.934371 1.934412 1.93445230.2 1.934490 1.934527 1.934562 1.934595 1.934627 1.934658 1.934687 1.934714 1.934740 1.93476530.3 1.934788 1.934810 1.934831 1.934850 1.934869 1.934886 1.934902 1.934917 1.934930 1.93494330.4 1.934955 1.934966 1.934976 1.934985 1.934993 1.935000 1.935007 1.935013 1.935018 1.93502330.5 1.935027 1.935031 1.935033 1.935036 1.935038 1.935040 1.935041 1.935042 1.935043 1.93504330.6 1.935044 1.935044 1.935044 1.935044 1.935044 1.935044 1.935044 1.935044 1.935045 1.93504530.7 1.935046 1.935047 1.935048 1.935050 1.935052 1.935055 1.935057 1.935061 1.935065 1.93507030.8 1.935075 1.935081 1.935088 1.935095 1.935103 1.935112 1.935122 1.935133 1.935144 1.93515730.9 1.935171 1.935185 1.935201 1.935218 1.935236 1.935255 1.935276 1.935297 1.935320 1.93534531.0 1.935370 1.935397 1.935425 1.935455 1.935486 1.935519 1.935553 1.935589 1.935626 1.93566531.1 1.935705 1.935747 1.935791 1.935836 1.935883 1.935932 1.935983 1.936035 1.936089 1.93614531.2 1.936202 1.936262 1.936323 1.936386 1.936452 1.936519 1.936587 1.936658 1.936731 1.93680531.3 1.936882 1.936960 1.937041 1.937123 1.937208 1.937294 1.937383 1.937473 1.937565 1.93766031.4 1.937756 1.937854 1.937955 1.938057 1.938161 1.938267 1.938376 1.938486 1.938598 1.93871231.5 1.938829 1.938947 1.939067 1.939189 1.939313 1.939439 1.939566 1.939696 1.939828 1.93996131.6 1.940096 1.940234 1.940373 1.940513 1.940656 1.940800 1.940946 1.941094 1.941244 1.94139531.7 1.941548 1.941702 1.941859 1.942017 1.942176 1.942337 1.942499 1.942663 1.942829 1.94299631.8 1.943164 1.943334 1.943505 1.943677 1.943851 1.944026 1.944203 1.944380 1.944559 1.94473931.9 1.944920 1.945102 1.945285 1.945469 1.945654 1.945840 1.946027 1.946215 1.946404 1.94659332.0 1.946783 1.946974 1.947166 1.947359 1.947552 1.947745 1.947939 1.948134 1.948329 1.94852532.1 1.948720 1.948917 1.949113 1.949310 1.949507 1.949705 1.949902 1.950100 1.950297 1.95049532.2 1.950693 1.950890 1.951088 1.951285 1.951483 1.951680 1.951877 1.952073 1.952269 1.95246532.3 1.952661 1.952856 1.953051 1.953245 1.953438 1.953631 1.953824 1.954015 1.954206 1.95439732.4 1.954586 1.954775 1.954963 1.955150 1.955336 1.955521 1.955705 1.955888 1.956070 1.95625132.5 1.956431 1.956610 1.956787 1.956964 1.957139 1.957313 1.957485 1.957656 1.957826 1.95799532.6 1.958162 1.958328 1.958492 1.958654 1.958816 1.958975 1.959133 1.959290 1.959445 1.95959832.7 1.959749 1.959899 1.960048 1.960194 1.960339 1.960482 1.960623 1.960763 1.960900 1.96103632.8 1.961170 1.961302 1.961433 1.961561 1.961688 1.961813 1.961935 1.962056 1.962175 1.96229232.9 1.962407 1.962521 1.962632 1.962741 1.962849 1.962954 1.963057 1.963159 1.963258 1.96335633.0 1.963452 1.963545 1.963637 1.963727 1.963814 1.963900 1.963984 1.964066 1.964146 1.96422433.1 1.964300 1.964375 1.964447 1.964518 1.964586 1.964653 1.964718 1.964781 1.964843 1.96490233.2 1.964960 1.965016 1.965070 1.965123 1.965173 1.965223 1.965270 1.965316 1.965360 1.96540233.3 1.965443 1.965483 1.965520 1.965557 1.965591 1.965625 1.965657 1.965687 1.965716 1.96574433.4 1.965770 1.965795 1.965819 1.965841 1.965863 1.965883 1.965902 1.965919 1.965936 1.96595233.5 1.965966 1.965980 1.965993 1.966004 1.966015 1.966025 1.966034 1.966042 1.966050 1.96605733.6 1.966063 1.966068 1.966073 1.966078 1.966081 1.966084 1.966087 1.966090 1.966091 1.96609333.7 1.966094 1.966095 1.966096 1.966096 1.966097 1.966097 1.966097 1.966097 1.966097 1.96609733.8 1.966097 1.966097 1.966098 1.966098 1.966099 1.966100 1.966101 1.966102 1.966104 1.96610633.9 1.966109 1.966112 1.966116 1.966120 1.966125 1.966130 1.966136 1.966142 1.966150 1.96615834.0 1.966167 1.966176 1.966187 1.966198 1.966211 1.966224 1.966238 1.966253 1.966269 1.96628734.1 1.966305 1.966325 1.966345 1.966367 1.966390 1.966414 1.966440 1.966467 1.966495 1.96652434.2 1.966555 1.966588 1.966621 1.966656 1.966693 1.966731 1.966770 1.966811 1.966854 1.96689834.3 1.966944 1.966991 1.967040 1.967090 1.967143 1.967196 1.967252 1.967309 1.967368 1.96742934.4 1.967491 1.967555 1.967621 1.967689 1.967758 1.967829 1.967902 1.967977 1.968053 1.96813134.5 1.968212 1.968293 1.968377 1.968463 1.968550 1.968639 1.968730 1.968823 1.968918 1.96901434.6 1.969112 1.969212 1.969314 1.969418 1.969523 1.969630 1.969739 1.969850 1.969963 1.97007734.7 1.970193 1.970311 1.970430 1.970551 1.970674 1.970799 1.970925 1.971053 1.971182 1.97131334.8 1.971446 1.971580 1.971716 1.971854 1.971993 1.972133 1.972275 1.972418 1.972563 1.97271034.9 1.972857 1.973006 1.973157 1.973309 1.973462 1.973616 1.973772 1.973928 1.974086 1.97424635.0 1.974406 1.974567 1.974730 1.974893 1.975058 1.975224 1.975390 1.975558 1.975726 1.97589635.1 1.976066 1.976237 1.976408 1.976581 1.976754 1.976928 1.977102 1.977278 1.977453 1.97763035.2 1.977806 1.977983 1.978161 1.978339 1.978518 1.978696 1.978875 1.979055 1.979234 1.97941435.3 1.979594 1.979774 1.979954 1.980134 1.980314 1.980494 1.980674 1.980854 1.981034 1.98121435.4 1.981393 1.981572 1.981751 1.981930 1.982108 1.982286 1.982463 1.982640 1.982817 1.98299335.5 1.983168 1.983343 1.983517 1.983691 1.983863 1.984036 1.984207 1.984377 1.984547 1.98471635.6 1.984884 1.985051 1.985217 1.985383 1.985547 1.985710 1.985872 1.986033 1.986193 1.98635135.7 1.986509 1.986665 1.986820 1.986974 1.987127 1.987278 1.987428 1.987577 1.987724 1.98787035.8 1.988014 1.988157 1.988298 1.988438 1.988577 1.988714 1.988849 1.988983 1.989115 1.98924635.9 1.989375 1.989503 1.989629 1.989753 1.989875 1.989996 1.990115 1.990233 1.990349 1.990463

31

Θ(x):

x 0 1 2 3 4 5 6 7 8 936.0 1.990575 1.990686 1.990794 1.990901 1.991007 1.991110 1.991212 1.991312 1.991410 1.99150736.1 1.991601 1.991694 1.991785 1.991875 1.991962 1.992048 1.992132 1.992214 1.992294 1.99237336.2 1.992450 1.992525 1.992598 1.992670 1.992740 1.992808 1.992874 1.992939 1.993002 1.99306336.3 1.993123 1.993180 1.993237 1.993291 1.993344 1.993396 1.993445 1.993493 1.993540 1.99358536.4 1.993629 1.993671 1.993711 1.993750 1.993788 1.993824 1.993859 1.993892 1.993924 1.99395436.5 1.993984 1.994012 1.994038 1.994064 1.994088 1.994111 1.994133 1.994154 1.994173 1.99419236.6 1.994209 1.994226 1.994241 1.994255 1.994269 1.994281 1.994293 1.994304 1.994314 1.99432336.7 1.994331 1.994339 1.994346 1.994352 1.994358 1.994363 1.994368 1.994372 1.994375 1.99437836.8 1.994381 1.994383 1.994385 1.994386 1.994387 1.994388 1.994389 1.994389 1.994390 1.99439036.9 1.994390 1.994390 1.994390 1.994390 1.994390 1.994390 1.994390 1.994391 1.994391 1.99439237.0 1.994393 1.994395 1.994396 1.994398 1.994401 1.994403 1.994407 1.994410 1.994415 1.99441937.1 1.994425 1.994431 1.994437 1.994445 1.994453 1.994462 1.994471 1.994481 1.994492 1.99450437.2 1.994517 1.994531 1.994546 1.994562 1.994578 1.994596 1.994615 1.994635 1.994656 1.99467837.3 1.994701 1.994725 1.994751 1.994778 1.994806 1.994835 1.994866 1.994898 1.994931 1.99496637.4 1.995002 1.995039 1.995078 1.995118 1.995160 1.995203 1.995248 1.995294 1.995341 1.99539137.5 1.995441 1.995494 1.995547 1.995603 1.995660 1.995718 1.995778 1.995840 1.995903 1.99596837.6 1.996035 1.996103 1.996173 1.996245 1.996318 1.996393 1.996469 1.996547 1.996627 1.99670937.7 1.996792 1.996877 1.996963 1.997051 1.997141 1.997233 1.997326 1.997421 1.997517 1.99761537.8 1.997715 1.997816 1.997919 1.998024 1.998130 1.998238 1.998347 1.998458 1.998570 1.99868437.9 1.998800 1.998917 1.999036 1.999156 1.999277 1.999400 1.999525 1.999651 1.999778 1.99990738.0 2.000037 2.000168 2.000301 2.000435 2.000570 2.000707 2.000844 2.000984 2.001124 2.00126538.1 2.001408 2.001552 2.001697 2.001842 2.001989 2.002137 2.002287 2.002437 2.002587 2.00273938.2 2.002892 2.003046 2.003200 2.003356 2.003512 2.003668 2.003826 2.003984 2.004143 2.00430338.3 2.004463 2.004623 2.004785 2.004946 2.005109 2.005271 2.005434 2.005598 2.005762 2.00592638.4 2.006090 2.006255 2.006419 2.006584 2.006750 2.006915 2.007080 2.007246 2.007411 2.00757738.5 2.007742 2.007907 2.008072 2.008237 2.008402 2.008567 2.008731 2.008895 2.009059 2.00922238.6 2.009385 2.009548 2.009710 2.009872 2.010033 2.010194 2.010354 2.010513 2.010672 2.01083138.7 2.010988 2.011145 2.011301 2.011456 2.011611 2.011764 2.011917 2.012069 2.012220 2.01237038.8 2.012519 2.012667 2.012814 2.012960 2.013105 2.013249 2.013392 2.013533 2.013674 2.01381338.9 2.013951 2.014088 2.014223 2.014357 2.014490 2.014622 2.014752 2.014881 2.015009 2.01513539.0 2.015259 2.015383 2.015505 2.015625 2.015744 2.015861 2.015977 2.016092 2.016205 2.01631639.1 2.016426 2.016534 2.016641 2.016746 2.016849 2.016951 2.017051 2.017150 2.017247 2.01734339.2 2.017437 2.017529 2.017619 2.017708 2.017796 2.017881 2.017965 2.018048 2.018128 2.01820739.3 2.018285 2.018361 2.018435 2.018507 2.018578 2.018648 2.018715 2.018781 2.018846 2.01890939.4 2.018970 2.019030 2.019088 2.019144 2.019199 2.019253 2.019305 2.019355 2.019404 2.01945239.5 2.019498 2.019542 2.019585 2.019627 2.019667 2.019706 2.019744 2.019780 2.019814 2.01984839.6 2.019880 2.019911 2.019940 2.019969 2.019996 2.020022 2.020047 2.020070 2.020093 2.02011439.7 2.020134 2.020153 2.020172 2.020189 2.020205 2.020220 2.020234 2.020248 2.020260 2.02027239.8 2.020283 2.020293 2.020302 2.020311 2.020319 2.020326 2.020332 2.020338 2.020344 2.02034839.9 2.020353 2.020356 2.020360 2.020362 2.020365 2.020367 2.020368 2.020370 2.020371 2.02037240.0 2.020372 2.020373 2.020373 2.020373 2.020374 2.020374 2.020374 2.020374 2.020374 2.02037440.1 2.020374 2.020374 2.020375 2.020375 2.020376 2.020378 2.020379 2.020381 2.020383 2.02038640.2 2.020388 2.020392 2.020396 2.020400 2.020405 2.020410 2.020416 2.020423 2.020430 2.02043840.3 2.020447 2.020456 2.020467 2.020478 2.020489 2.020502 2.020515 2.020530 2.020545 2.02056140.4 2.020579 2.020597 2.020616 2.020636 2.020657 2.020680 2.020703 2.020728 2.020754 2.02078140.5 2.020809 2.020838 2.020869 2.020900 2.020933 2.020968 2.021003 2.021040 2.021079 2.02111840.6 2.021159 2.021202 2.021245 2.021290 2.021337 2.021385 2.021434 2.021485 2.021538 2.02159140.7 2.021647 2.021703 2.021762 2.021822 2.021883 2.021946 2.022010 2.022076 2.022143 2.02221240.8 2.022282 2.022354 2.022428 2.022503 2.022579 2.022658 2.022737 2.022818 2.022901 2.02298540.9 2.023071 2.023159 2.023247 2.023338 2.023430 2.023523 2.023618 2.023714 2.023812 2.02391141.0 2.024012 2.024114 2.024218 2.024323 2.024430 2.024537 2.024647 2.024757 2.024870 2.02498341.1 2.025098 2.025214 2.025331 2.025450 2.025570 2.025691 2.025813 2.025937 2.026062 2.02618841.2 2.026315 2.026443 2.026573 2.026703 2.026835 2.026968 2.027101 2.027236 2.027372 2.02750841.3 2.027646 2.027784 2.027924 2.028064 2.028205 2.028347 2.028490 2.028633 2.028777 2.02892241.4 2.029068 2.029214 2.029360 2.029508 2.029656 2.029804 2.029953 2.030102 2.030252 2.03040341.5 2.030553 2.030704 2.030856 2.031007 2.031159 2.031311 2.031464 2.031616 2.031769 2.03192241.6 2.032075 2.032227 2.032380 2.032533 2.032686 2.032839 2.032992 2.033144 2.033297 2.03344941.7 2.033601 2.033753 2.033904 2.034055 2.034206 2.034357 2.034507 2.034656 2.034805 2.03495441.8 2.035102 2.035250 2.035397 2.035543 2.035689 2.035834 2.035979 2.036123 2.036266 2.03640841.9 2.036549 2.036690 2.036830 2.036969 2.037107 2.037244 2.037380 2.037515 2.037649 2.037783

32

Θ(x):

x 0 1 2 3 4 5 6 7 8 942.0 2.037915 2.038046 2.038176 2.038305 2.038433 2.038559 2.038685 2.038809 2.038932 2.03905442.1 2.039175 2.039294 2.039413 2.039530 2.039645 2.039759 2.039872 2.039984 2.040094 2.04020342.2 2.040311 2.040417 2.040521 2.040625 2.040726 2.040827 2.040926 2.041023 2.041119 2.04121442.3 2.041307 2.041398 2.041489 2.041577 2.041664 2.041750 2.041834 2.041916 2.041997 2.04207742.4 2.042155 2.042231 2.042306 2.042380 2.042452 2.042522 2.042591 2.042658 2.042724 2.04278942.5 2.042852 2.042913 2.042973 2.043031 2.043088 2.043144 2.043198 2.043250 2.043302 2.04335142.6 2.043400 2.043447 2.043492 2.043536 2.043579 2.043620 2.043661 2.043699 2.043737 2.04377342.7 2.043808 2.043841 2.043874 2.043905 2.043935 2.043964 2.043991 2.044018 2.044043 2.04406742.8 2.044090 2.044112 2.044133 2.044153 2.044172 2.044190 2.044207 2.044223 2.044238 2.04425242.9 2.044266 2.044278 2.044290 2.044301 2.044311 2.044320 2.044329 2.044337 2.044344 2.04435143.0 2.044357 2.044363 2.044368 2.044372 2.044376 2.044380 2.044383 2.044385 2.044388 2.04439043.1 2.044391 2.044393 2.044394 2.044395 2.044395 2.044396 2.044396 2.044396 2.044396 2.04439643.2 2.044396 2.044396 2.044396 2.044396 2.044397 2.044397 2.044397 2.044398 2.044399 2.04440043.3 2.044401 2.044403 2.044405 2.044407 2.044410 2.044413 2.044416 2.044420 2.044425 2.04443043.4 2.044435 2.044441 2.044448 2.044455 2.044463 2.044472 2.044481 2.044491 2.044502 2.04451443.5 2.044526 2.044539 2.044553 2.044568 2.044584 2.044601 2.044619 2.044637 2.044657 2.04467743.6 2.044699 2.044722 2.044746 2.044770 2.044796 2.044823 2.044852 2.044881 2.044911 2.04494343.7 2.044976 2.045010 2.045045 2.045082 2.045120 2.045159 2.045200 2.045241 2.045284 2.04532943.8 2.045374 2.045421 2.045470 2.045520 2.045571 2.045623 2.045677 2.045732 2.045789 2.04584743.9 2.045907 2.045968 2.046030 2.046094 2.046159 2.046226 2.046294 2.046363 2.046434 2.04650644.0 2.046580 2.046655 2.046732 2.046810 2.046889 2.046970 2.047053 2.047136 2.047222 2.04730844.1 2.047396 2.047485 2.047576 2.047668 2.047761 2.047856 2.047952 2.048050 2.048148 2.04824944.2 2.048350 2.048453 2.048557 2.048662 2.048768 2.048876 2.048985 2.049095 2.049206 2.04931844.3 2.049432 2.049547 2.049663 2.049780 2.049898 2.050017 2.050137 2.050258 2.050380 2.05050344.4 2.050627 2.050752 2.050878 2.051005 2.051133 2.051262 2.051391 2.051521 2.051652 2.05178444.5 2.051916 2.052050 2.052183 2.052318 2.052453 2.052589 2.052725 2.052862 2.053000 2.05313744.6 2.053276 2.053415 2.053554 2.053694 2.053834 2.053974 2.054115 2.054256 2.054397 2.05453844.7 2.054680 2.054822 2.054964 2.055106 2.055248 2.055390 2.055532 2.055674 2.055816 2.05595944.8 2.056101 2.056242 2.056384 2.056526 2.056667 2.056808 2.056949 2.057090 2.057230 2.05737044.9 2.057510 2.057649 2.057788 2.057926 2.058064 2.058201 2.058338 2.058475 2.058610 2.05874545.0 2.058880 2.059014 2.059147 2.059279 2.059411 2.059542 2.059672 2.059801 2.059930 2.06005745.1 2.060184 2.060310 2.060435 2.060559 2.060682 2.060805 2.060926 2.061046 2.061165 2.06128345.2 2.061400 2.061516 2.061630 2.061744 2.061857 2.061968 2.062078 2.062187 2.062295 2.06240145.3 2.062507 2.062611 2.062713 2.062815 2.062915 2.063014 2.063112 2.063208 2.063303 2.06339745.4 2.063489 2.063580 2.063669 2.063757 2.063844 2.063930 2.064014 2.064096 2.064178 2.06425845.5 2.064336 2.064413 2.064489 2.064563 2.064636 2.064707 2.064777 2.064846 2.064913 2.06497945.6 2.065043 2.065106 2.065167 2.065228 2.065286 2.065344 2.065400 2.065454 2.065508 2.06555945.7 2.065610 2.065659 2.065707 2.065753 2.065799 2.065842 2.065885 2.065926 2.065966 2.06600545.8 2.066043 2.066079 2.066114 2.066148 2.066180 2.066212 2.066242 2.066271 2.066299 2.06632645.9 2.066352 2.066377 2.066401 2.066423 2.066445 2.066466 2.066485 2.066504 2.066522 2.06653946.0 2.066554 2.066570 2.066584 2.066597 2.066610 2.066621 2.066632 2.066643 2.066652 2.06666146.1 2.066669 2.066677 2.066684 2.066690 2.066696 2.066701 2.066706 2.066710 2.066714 2.06671746.2 2.066720 2.066723 2.066725 2.066727 2.066728 2.066730 2.066731 2.066732 2.066732 2.06673346.3 2.066733 2.066733 2.066733 2.066733 2.066733 2.066733 2.066733 2.066733 2.066733 2.06673446.4 2.066734 2.066735 2.066735 2.066736 2.066738 2.066739 2.066741 2.066743 2.066745 2.06674846.5 2.066751 2.066755 2.066759 2.066764 2.066769 2.066774 2.066781 2.066787 2.066795 2.06680346.6 2.066811 2.066820 2.066830 2.066841 2.066853 2.066865 2.066878 2.066892 2.066906 2.06692246.7 2.066938 2.066955 2.066974 2.066993 2.067013 2.067034 2.067056 2.067079 2.067103 2.06712846.8 2.067154 2.067181 2.067210 2.067239 2.067270 2.067301 2.067334 2.067368 2.067403 2.06744046.9 2.067477 2.067516 2.067556 2.067597 2.067640 2.067683 2.067728 2.067775 2.067822 2.06787147.0 2.067921 2.067972 2.068025 2.068079 2.068135 2.068191 2.068249 2.068309 2.068369 2.06843147.1 2.068495 2.068559 2.068625 2.068693 2.068761 2.068831 2.068903 2.068975 2.069049 2.06912547.2 2.069201 2.069279 2.069358 2.069439 2.069521 2.069604 2.069689 2.069774 2.069861 2.06995047.3 2.070039 2.070130 2.070222 2.070315 2.070410 2.070505 2.070602 2.070700 2.070800 2.07090047.4 2.071001 2.071104 2.071208 2.071313 2.071419 2.071526 2.071634 2.071743 2.071853 2.07196447.5 2.072076 2.072189 2.072303 2.072418 2.072533 2.072650 2.072768 2.072886 2.073005 2.07312547.6 2.073246 2.073367 2.073490 2.073613 2.073736 2.073861 2.073986 2.074111 2.074237 2.07436447.7 2.074491 2.074619 2.074747 2.074876 2.075005 2.075135 2.075265 2.075396 2.075526 2.07565747.8 2.075789 2.075920 2.076052 2.076184 2.076317 2.076449 2.076582 2.076714 2.076847 2.07698047.9 2.077113 2.077246 2.077379 2.077512 2.077644 2.077777 2.077909 2.078042 2.078174 2.078306

33

x 0 1 2 3 4 5 6 7 8 948.0 2.078438 2.078569 2.078700 2.078831 2.078962 2.079092 2.079222 2.079351 2.079480 2.07960848.1 2.079736 2.079864 2.079991 2.080117 2.080243 2.080368 2.080492 2.080616 2.080740 2.08086248.2 2.080984 2.081105 2.081225 2.081345 2.081463 2.081581 2.081698 2.081814 2.081929 2.08204448.3 2.082157 2.082270 2.082381 2.082492 2.082601 2.082710 2.082817 2.082923 2.083029 2.08313348.4 2.083236 2.083338 2.083439 2.083539 2.083637 2.083735 2.083831 2.083926 2.084020 2.08411348.5 2.084204 2.084295 2.084384 2.084471 2.084558 2.084643 2.084727 2.084810 2.084891 2.08497148.6 2.085050 2.085127 2.085204 2.085279 2.085352 2.085424 2.085495 2.085565 2.085633 2.08570048.7 2.085766 2.085830 2.085893 2.085955 2.086015 2.086074 2.086132 2.086189 2.086244 2.08629848.8 2.086350 2.086401 2.086451 2.086500 2.086548 2.086594 2.086639 2.086682 2.086725 2.08676648.9 2.086806 2.086845 2.086883 2.086919 2.086954 2.086988 2.087021 2.087053 2.087084 2.08711349.0 2.087142 2.087169 2.087196 2.087221 2.087246 2.087269 2.087291 2.087312 2.087333 2.08735249.1 2.087371 2.087388 2.087405 2.087421 2.087436 2.087450 2.087464 2.087476 2.087488 2.08749949.2 2.087510 2.087520 2.087529 2.087537 2.087545 2.087552 2.087558 2.087564 2.087570 2.08757549.3 2.087579 2.087583 2.087587 2.087590 2.087593 2.087595 2.087598 2.087599 2.087601 2.08760249.4 2.087603 2.087604 2.087604 2.087605 2.087605 2.087605 2.087605 2.087605 2.087605 2.08760549.5 2.087605 2.087606 2.087606 2.087606 2.087606 2.087607 2.087607 2.087608 2.087609 2.08761149.6 2.087612 2.087614 2.087617 2.087619 2.087622 2.087625 2.087629 2.087633 2.087638 2.08764349.7 2.087649 2.087655 2.087662 2.087669 2.087677 2.087686 2.087695 2.087705 2.087716 2.08772749.8 2.087739 2.087752 2.087766 2.087780 2.087795 2.087811 2.087828 2.087846 2.087865 2.08788449.9 2.087905 2.087926 2.087949 2.087972 2.087996 2.088022 2.088048 2.088076 2.088104 2.088134x 0 1 2 3 4 5 6 7 8 950 2.088164 2.088532 2.089018 2.089629 2.090364 2.091220 2.092185 2.093248 2.094389 2.09558951 2.096824 2.098070 2.099302 2.100496 2.101629 2.102681 2.103635 2.104478 2.105201 2.10580152 2.106279 2.106640 2.106895 2.107058 2.107148 2.107186 2.107193 2.107195 2.107215 2.10727753 2.107401 2.107606 2.107908 2.108319 2.108845 2.109489 2.110248 2.111115 2.112080 2.11312654 2.114236 2.115388 2.116560 2.117729 2.118872 2.119965 2.120991 2.121930 2.122769 2.12349855 2.124112 2.124610 2.124995 2.125276 2.125464 2.125575 2.125629 2.125645 2.125646 2.12565556 2.125694 2.125784 2.125945 2.126191 2.126535 2.126986 2.127548 2.128220 2.128998 2.12987357 2.130831 2.131858 2.132932 2.134035 2.135144 2.136237 2.137292 2.138291 2.139214 2.14004858 2.140782 2.141409 2.141925 2.142333 2.142638 2.142851 2.142984 2.143055 2.143083 2.14308759 2.143090 2.143113 2.143176 2.143298 2.143496 2.143782 2.144167 2.144655 2.145249 2.14594660 2.146738 2.147615 2.148563 2.149565 2.150602 2.151653 2.152698 2.153716 2.154687 2.15559561 2.156422 2.157159 2.157796 2.158329 2.158759 2.159088 2.159325 2.159481 2.159571 2.15961162 2.159622 2.159623 2.159634 2.159676 2.159767 2.159923 2.160159 2.160485 2.160907 2.16143063 2.162052 2.162768 2.163569 2.164444 2.165377 2.166351 2.167347 2.168346 2.169326 2.17027164 2.171160 2.171980 2.172718 2.173364 2.173913 2.174362 2.174714 2.174974 2.175153 2.17526365 2.175319 2.175339 2.175341 2.175346 2.175371 2.175436 2.175557 2.175749 2.176022 2.17638666 2.176844 2.177398 2.178043 2.178774 2.179580 2.180448 2.181362 2.182305 2.183258 2.18420167 2.185118 2.185989 2.186800 2.187537 2.188190 2.188752 2.189220 2.189593 2.189877 2.19007968 2.190209 2.190282 2.190313 2.190320 2.190321 2.190335 2.190380 2.190471 2.190625 2.19085269 2.191163 2.191562 2.192053 2.192633 2.193298 2.194040 2.194845 2.195702 2.196593 2.19750170 2.198408 2.199297 2.200149 2.200949 2.201684 2.202342 2.202916 2.203400 2.203794 2.20410071 2.204324 2.204476 2.204567 2.204611 2.204625 2.204626 2.204632 2.204661 2.204728 2.20484872 2.205035 2.205298 2.205645 2.206078 2.206598 2.207201 2.207881 2.208628 2.209429 2.21027073 2.211134 2.212005 2.212865 2.213697 2.214485 2.215216 2.215878 2.216461 2.216961 2.21737474 2.217701 2.217948 2.218120 2.218230 2.218289 2.218313 2.218317 2.218319 2.218335 2.21838275 2.218474 2.218626 2.218847 2.219145 2.219526 2.219990 2.220535 2.221157 2.221848 2.22259676 2.223388 2.224209 2.225043 2.225874 2.226685 2.227460 2.228185 2.228848 2.229440 2.22995377 2.230384 2.230732 2.231000 2.231194 2.231323 2.231398 2.231433 2.231443 2.231443 2.23145178 2.231482 2.231551 2.231672 2.231855 2.232110 2.232442 2.232854 2.233346 2.233913 2.23455079 2.235247 2.235992 2.236770 2.237568 2.238369 2.239158 2.239918 2.240636 2.241300 2.24189880 2.242422 2.242869 2.243236 2.243525 2.243740 2.243889 2.243981 2.244029 2.244047 2.24404981 2.244052 2.244071 2.244120 2.244214 2.244364 2.244580 2.244868 2.245232 2.245673 2.24618982 2.246775 2.247423 2.248121 2.248858 2.249620 2.250391 2.251156 2.251901 2.252610 2.25327283 2.253874 2.254409 2.254870 2.255256 2.255564 2.255800 2.255969 2.256079 2.256141 2.25616884 2.256175 2.256176 2.256186 2.256219 2.256291 2.256411 2.256592 2.256839 2.257159 2.25755385 2.258021 2.258558 2.259158 2.259812 2.260508 2.261233 2.261974 2.262716 2.263443 2.26414286 2.264800 2.265405 2.265948 2.266423 2.266825 2.267153 2.267409 2.267597 2.267725 2.26780387 2.267842 2.267855 2.267856 2.267860 2.267882 2.267934 2.268029 2.268178 2.268389 2.26866888 2.269018 2.269440 2.269931 2.270485 2.271096 2.271751 2.272441 2.273151 2.273868 2.27457789 2.275264 2.275917 2.276523 2.277073 2.277559 2.277976 2.278322 2.278597 2.278806 2.27895390 2.279047 2.279098 2.279119 2.279124 2.279125 2.279137 2.279173 2.279247 2.279368 2.27954691 2.279787 2.280097 2.280476 2.280922 2.281433 2.282001 2.282617 2.283271 2.283950 2.28464192 2.285331 2.286005 2.286651 2.287256 2.287811 2.288307 2.288738 2.289101 2.289395 2.28962393 2.289788 2.289899 2.289965 2.289996 2.290005 2.290006 2.290012 2.290036 2.290090 2.29018794 2.290335 2.290543 2.290814 2.291153 2.291557 2.292026 2.292553 2.293131 2.293749 2.29439795 2.295062 2.295731 2.296391 2.297028 2.297631 2.298189 2.298693 2.299137 2.299515 2.29982796 2.300074 2.300258 2.300387 2.300467 2.300510 2.300526 2.300529 2.300531 2.300545 2.30058497 2.300659 2.300781 2.300957 2.301194 2.301494 2.301859 2.302288 2.302775 2.303315 2.30389898 2.304515 2.305153 2.305801 2.306445 2.307072 2.307671 2.308231 2.308741 2.309196 2.30958999 2.309918 2.310183 2.310386 2.310532 2.310628 2.310683 2.310708 2.310714 2.310715 2.310722

Θ(100) = 2.310 749

34

c) Integrals:

Holds ∫J 2

0 (x) dx = x[J 20 (x) + J 2

1 (x)] +∫

J 21 (x) dx ,∫

I 20 (x) dx = x[I 2

0 (x)− I 21 (x)]−

∫I 21 (x) dx .

This formulas express every integral by the other one. Therefore the next integrals are given with∫

J20 (x) dx

or∫

I20 (x) dx . These are represented by the functions Θ(x) respectively Ω(x), see page 22 .

Furthermore, one has∫x2 J2

0 (x) dx =18[(2 x3 + x) J2

0 (x) + 2x2J0(x)J1(x) + 2x3 J21 (x)

]− 1

8

∫J2

0 (x) dx

∫x2 I2

0 (x) dx =18[(2 x3 − x) I2

0 (x) + 2x2I0(x)I1(x)− 2x3 I21 (x)

]+

18

∫I20 (x) dx∫

x2 J21 (x) dx =

18[(2 x3 − 3x) J2

0 (x)− 6x2J0(x)J1(x) + 2x3 J21 (x)

]+

38

∫J2

0 (x) dx∫x2 I2

1 (x) dx =18[(−2 x3 − 3x) I2

0 (x) + 6x2I0(x)I1(x) + 2x3 I21 (x)

]+

38

∫I20 (x) dx∫

x4 J20 (x) dx =

=1

128[(16 x5 + 18 x3 − 27 x) J2

0 (x) + (48 x4 − 54 x2)J0(x)J1(x) + (16 x5 − 54 x3) J21 (x)

]+

27128

∫J2

0 (x) dx∫x4 I2

0 (x) dx =

=1

128[(16 x5 − 18 x3 − 27 x) I2

0 (x) + (48 x4 + 54 x2)I0(x)I1(x) + (−16 x5 − 54 x3) I21 (x)

]+

27128

∫I20 (x) dx∫

x4 J21 (x) dx =

=1

128[(16x5 − 30 x3 + 45 x) J2

0 (x) + (−80 x4 + 90 x2)J0(x)J1(x) + (16 x5 + 90 x3) J21 (x)

]−

− 45128

∫J2

0 (x) dx∫x4 I2

1 (x) dx =

=1

128[(−16 x5 − 30 x3 − 45 x) I2

0 (x) + (80 x4 + 90 x2)I0(x)I1(x) + (16 x5 − 90 x3) I21 (x)

]+

+45128

∫I20 (x) dx

With ∫x2n J2

0 (x) dx =1βn

[An(x) J2

0 (x) + Bn(x)J0(x)J1(x) + Cn(x) J21 (x) + γn

∫J2

0 (x) dx

]∫

x2n I20 (x) dx =

1β∗n

[A∗n(x) I2

0 (x) + B∗n(x)I0(x)I1(x) + C∗n(x) I21 (x) + γ∗n

∫I20 (x) dx

]∫

x2n J21 (x) dx =

1ξn

[Pn(x) J2

0 (x) + Qn(x)J0(x)J1(x) + Rn(x) J21 (x) + %n

∫J2

0 (x) dx

]∫

x2n I21 (x) dx =

1ξ∗n

[P ∗n(x) I2

0 (x) + Q∗n(x)I0(x)I1(x) + R∗n(x) I21 (x) + %∗n

∫I20 (x) dx

]

35

holdsβ3 = 3072 , γ3 = −3375

A3(x) = 256 x7 + 1200 x5 − 2250 x3 + 3375 x

B3(x) = 1280 x6 − 6000 x4 + 6750 x2 , C3(x) = 256 x7 − 2000 x5 + 6750 x3

β∗3 = 3072 , γ∗3 = 3375

A∗3(x) = 256 x7 − 1200 x5 − 2250 x3 − 3375 x

B∗3(x) = 1280 x6 + 6000 x4 + 6750 x2 , C∗3 (x) = −256 x7 − 2000 x5 − 6750 x3

ξ3 = 3072 , %3 = 4725

P3(x) = 256 x7 − 1680 x5 + 3150 x3 − 4725 x

Q3(x) = −1792 x6 + 8400 x4 − 9450 x2 , R3(x) = 256 x7 + 2800 x5 − 9450 x3

ξ∗3 = 3072 , %∗3 = 4725

P ∗3 (x) = −256 x7 − 1680 x5 − 3150 x3 − 4725 x

Q∗3(x) = 1792 x6 + 8400 x4 + 9450 x2 , R∗3(x) = 256 x7 − 2800 x5 − 9450 x3

β4 = 98304 , γ4 = 1157625

A4(x) = 6144 x9 + 62720x7 − 411600 x5 + 771750 x3 − 1157625x

B4(x) = 43008 x8 − 439040 x6 + 2058000x4 − 2315250 x2

C4(x) = 6144 x9 − 87808 x7 + 686000 x5 − 2315250 x3

β∗4 = 98304 , γ∗4 = 1157625

A∗4(x) = 6144 x9 − 62720 x7 − 411600 x5 − 771750 x3 − 1157625 x

B∗4(x) = 43008 x8 + 439040 x6 + 2058000 x4 + 2315250x2

C∗4 (x) = −6144 x9 − 87808 x7 − 686000 x5 − 2315250 x3

ξ4 = 32768 , %4 = −496125

P4(x) = 2048 x9 − 26880 x7 + 176400 x5 − 330750 x3 + 496125 x

Q4(x) = −18432 x8 + 188160 x6 − 882000 x4 + 992250 x2

R4(x) = 2048 x9 + 37632 x7 − 294000 x5 + 992250 x3

ξ∗4 = 32768 , %∗4 = 496125

P ∗4 (x) = −2048 x9 − 26880 x7 − 176400 x5 − 330750 x3 − 496125 x

Q∗4(x) = 18432 x8 + 188160 x6 + 882000 x4 + 992250 x2

R∗4(x) = 2048 x9 − 37632 x7 − 294000 x5 − 992250 x3

β5 = 1310720 , γ5 = −281302875

A5(x) = 65536 x11 + 1161216x9 − 15240960 x7 + 100018800x5 − 187535250x3 + 281302875x

B5(x) = 589824 x10 − 10450944 x8 + 106686720x6 − 500094000x4 + 562605750x2

C5(x) = 65536 x11 − 1492992 x9 + 21337344 x7 − 166698000 x5 + 562605750x3

β∗5 = 1310720 , γ∗5 = 281302875∗

36

A5(x)∗ = 65536 x11 − 1161216x9 − 15240960 x7 − 100018800x5 − 187535250 x3 − 281302875 x

B5(x)∗ = 589824 x10 + 10450944 x8 + 106686720x6 + 500094000x4 + 562605750x2

C5(x)∗ = −65536 x11 − 1492992x9 − 21337344 x7 − 166698000x5 − 562605750 x3

ξ5 = 1310720 , %5 = 343814625

P5(x) = 65536 x11 − 1419264x9 + 18627840x7 − 122245200x5 + 229209750x3 − 343814625 x

Q5(x) = −720896 x10 + 12773376 x8 − 130394880x6 + 611226000x4 − 687629250x2

R5(x) = 65536 x11 + 1824768x9 − 26078976 x7 + 203742000x5 − 687629250x3

ξ∗5 = 1310720 , %∗5 = 343814625

P ∗5 (x) = −65536 x11 − 1419264 x9 − 18627840 x7 − 122245200x5 − 229209750x3 − 343814625x

Q∗5(x) = 720896 x10 + 12773376 x8 + 130394880x6 + 611226000x4 + 687629250x2

R∗5(x) = 65536 x11 − 1824768 x9 − 26078976 x7 − 203742000 x5 − 687629250x3

β6 = 62914560 , γ6 = 374414126625

A6(x) = 2621440 x13+71368704x11−1545578496 x9+20285717760x7−133125022800x5+249609417750x3−

−374414126625x

B6(x) = 28835840 x12−785055744x10+13910206464x8−142000024320 x6+665625114000x4−748828253250 x2

C6(x) = 2621440 x13−87228416 x11+1987172352x9−28400004864x7+221875038000x5−748828253250x3

β∗6 = 62914560 , γ∗6 = 374414126625

A∗6(x) = 2621440 x13−71368704 x11−1545578496 x9−20285717760x7−133125022800 x5−249609417750x3−

−374414126625 x

B∗6(x) = 28835840 x12+785055744x10+13910206464x8+142000024320x6+665625114000x4+748828253250x2

C∗6 (x) = −2621440 x13−87228416 x11−1987172352 x9−28400004864x7−221875038000 x5−748828253250x3

ξ6 = 62914560 , %6 = −442489422375

P6(x) = 2621440 x13 − 84344832 x11 + 1826592768 x9 − 23974030080x7 + 157329572400x5−

−294992948250 x3 + 442489422375x

Q6(x) = −34078720 x12 + 927793152x10 − 16439334912 x8 + 167818210560x6−

−786647862000x4 + 884978844750x2

R6(x) = 2621440 x13 + 103088128x11 − 2348476416 x9 + 33563642112x7−

−262215954000x5 + 884978844750x3

ξ∗6 = 62914560 , %∗6 = 442489422375

P ∗6 (x) = −2621440 x13 − 84344832 x11 − 1826592768x9 − 23974030080x7 − 157329572400x5−

−294992948250 x3 − 442489422375x

Q∗6(x) = 34078720 x12 + 927793152x10 + 16439334912x8 + 167818210560x6+

+786647862000x4 + 884978844750x2

R∗6(x) = 2621440 x13 − 103088128 x11 − 2348476416 x9 − 33563642112x7−

−262215954000x5 − 884978844750x3

37

β7 = 3523215360 , γ7 = −822587836195125

A7(x) = 125829120 x15 + 4873256960x13 − 156797042688 x11 + 3395635955712x9 − 44567721918720x7+

+292475675091600x5 − 548391890796750x3 + 822587836195125x

B7(x) = 1635778560 x14 − 63352340480x12 + 1724767469568x10 − 30560723601408x8+

+311974053431040x6 − 1462378375458000x4 + 1645175672390250x2

C7(x) = 125829120 x15 − 5759303680 x13 + 191640829952x11 − 4365817657344x9 + 62394810686208x7−

−487459458486000x5 + 1645175672390250x3

β∗7 = 3523215360 , γ∗7 = 822587836195125

A∗7(x) = 125829120 x15 − 4873256960 x13 − 156797042688x11 − 3395635955712x9 − 44567721918720x7−

−292475675091600x5 − 548391890796750x3 − 822587836195125x

B∗7(x) = 1635778560 x14 + 63352340480x12 + 1724767469568x10 + 30560723601408x8+

+311974053431040x6 + 1462378375458000x4 + 1645175672390250x2

C∗7 (x) = −125829120 x15− 5759303680 x13− 191640829952x11− 4365817657344x9− 62394810686208x7−

−487459458486000x5 − 1645175672390250x3

ξ7 = 234881024 , %7 = 63275987399625

P7(x) = 8388608 x15 − 374865920 x13 + 12061310976x11 − 261202765824x9 + 3428286301440x7−

−22498128853200x5 + 42183991599750x3 − 63275987399625x

Q7(x) = −125829120x14 + 4873256960x12− 132674420736x10 + 2350824892416 x8− 23998004110080x6+

+112490644266000x4 − 126551974799250x2

R7(x) = 8388608 x15 + 443023360x13 − 14741602304x11 + 335832127488x9 − 4799600822016x7+

+37496881422000x5 − 126551974799250x3

ξ∗7 = 234881024 , %∗7 = 63275987399625

P ∗7 (x) = −8388608 x15 − 374865920x13 − 12061310976x11 − 261202765824 x9 − 3428286301440x7−

−22498128853200x5 − 42183991599750x3 − 63275987399625x

Q∗7(x) = 125829120 x14 + 4873256960x12 + 132674420736x10 + 2350824892416x8 + 23998004110080x6+

+112490644266000 x4 + 126551974799250x2

R∗7(x) = 8388608 x15 − 443023360 x13 − 14741602304x11 − 335832127488x9 − 4799600822016x7−

−37496881422000x5 − 126551974799250x3

β8 = 15032385536 , γ8 = 185082263143903125

A8(x) = 469762048 x17 + 24536678400x15 − 1096482816000x13 + 35279334604800x11−

−764018090035200x9 + 10027737431712000x7 − 65807026895610000x5 + 123388175429268750x3−

−185082263143903125x

B8(x) = 7046430720 x16 − 368050176000x14 + 14254276608000x12 − 388072680652800x10+

+6876162810316800x8 − 70194162021984000x6 + 329035134478050000x4 − 370164526287806250x2

C8(x) = 469762048 x17 − 28311552000x15 + 1295843328000x13 − 43119186739200x11+

+982308972902400x9 − 14038832404396800x7 + 109678378159350000x5 − 370164526287806250x3

38

β∗8 = 15032385536 , γ∗8 = 185082263143903125

A∗8(x) = 469762048 x17 − 24536678400x15 − 1096482816000x13 − 35279334604800 x11−

−764018090035200x9 − 10027737431712000x7 − 65807026895610000x5 − 123388175429268750x3−

−185082263143903125x

B∗8(x) = 7046430720 x16 + 368050176000x14 + 14254276608000x12 + 388072680652800x10+

+6876162810316800x8 + 70194162021984000x6 + 329035134478050000x4 + 370164526287806250x2

C∗8 (x) = −469762048 x17 − 28311552000x15 − 1295843328000x13 − 43119186739200 x11−

−982308972902400x9 − 14038832404396800x7 − 109678378159350000x5 − 370164526287806250x3

ξ8 = 15032385536 , %8 = −209759898229756875

P8(x) = 469762048 x17−27808235520x15+1242680524800x13−39983245885440x11+865887168706560x9−

−11364769089273600x7 + 74581297148358000x5 − 139839932153171250x3 + 209759898229756875x

Q8(x) = −7985954816 x16 + 417123532800x14 − 16154846822400x12 + 439815704739840x10−

−7792984518359040x8 + 79553383624915200x6 − 372906485741790000x4 + 419519796459513750x2

R8(x) = 469762048 x17 + 32086425600x15 − 1468622438400x13 + 48868411637760x11−

−1113283502622720x9 + 15910676724983040x7 − 124302161913930000x5 + 419519796459513750x3

ξ∗8 = 15032385536 , %∗8 = 209759898229756875

P ∗8 (x) = −469762048 x17−27808235520x15−1242680524800x13−39983245885440x11−865887168706560x9−

−11364769089273600x7 − 74581297148358000x5 − 139839932153171250x3 − 209759898229756875x

Q∗8(x) = 7985954816 x16 + 417123532800x14 + 16154846822400x12 + 439815704739840x10+

+7792984518359040x8 + 79553383624915200x6 + 372906485741790000x4 + 419519796459513750x2

R∗8(x) = 469762048 x17 − 32086425600x15 − 1468622438400x13 − 48868411637760x11−

−1113283502622720x9 − 15910676724983040x7 − 124302161913930000x5 − 419519796459513750x3

39

XII. Integrals of the type∫

x2n+1Z0(x)Z1(x) dx

The integrals∫

J20 (x) dx and

∫I20 (x) dx may be defined as the functions Θ(x) and Ω(x) in XI, page 22 .∫xJ0(x)J1(x) dx = −1

2J2

0 (x) +12

∫J2

0 (x) dx

∫x I0(x)I1(x) dx =

12

I20 (x)− 1

2

∫I20 (x) dx∫

x3 J0(x)J1(x) dx =116[(−2 x3 + 3 x) J2

0 (x) + 6 x2J0(x)J1(x) + 6 x3 J21 (x)

]− 3

16

∫J2

0 (x) dx∫x3 I0(x)I1(x) dx =

116[(2 x3 + 3 x) I2

0 (x)− 6x2I0(x)I1(x) + x3 I21 (x)

]− 3

16

∫I20 (x) dx

With∫x2n+1 J0(x) · J1(x) dx =

1βn

[An(x) J2

0 (x) + Bn(x)J0(x)J1(x) + Cn(x) J21 (x) + γn

∫J2

0 (x) dx

]∫

x2n+1 I0(x) · I1(x) dx =1β∗n

[A∗n(x) I2

0 (x) + I∗n(x)I0(x)I1(x) + C∗n(x) I21 (x) + γ∗n

∫I20 (x) dx

]holds

β2 = 256 , γ2 = 135

A2(x) = −48 x5 + 90 x3 − 135 x , B2(x) = 240 x4 − 270 x2 , C2(x) = 80 x5 − 270 x3

β∗2 = 256 , γ∗2 = −135

A∗2(x) = 48 x5 + 90 x3 + 135 x , B∗2(x) = −240 x4 − 270 x2 , C∗2 (x) = 80 x5 + 270 x3

β3 = 30720 , γ3 = −118125

A3(x) = −6400 x7 + 42000 x5 − 78750 x3 + 118125 x

B3(x) = 44800 x6 − 210000 x4 + 236250 x2 , C3(x) = 8960 x7 − 70000 x5 + 236250 x3

β∗3 = 30720 , γ∗3 = −118125

A∗3(x) = 6400 x7 + 42000 x5 + 78750x3 + 118125 x

B∗3(x) = −44800 x6 − 210000 x4 − 236250 x2 , C∗3 (x) = 8960 x7 + 70000 x5 + 236250 x3

β4 = 983040 , γ4 = 52093125

A4(x) = −215040 x9 + 2822400 x7 − 18522000 x5 + 34728750x3 − 52093125 x

B4(x) = 1935360 x8 − 19756800 x6 + 92610000 x4 − 104186250x2

C4(x) = 276480 x9 − 3951360 x7 + 30870000x5 − 104186250 x3

β∗4 = 983040 , γ∗4 = −52093125

A∗4(x) = 215040 x9 + 2822400 x7 + 18522000x5 + 34728750x3 + 52093125 x

B∗4(x) = −1935360 x8 − 19756800 x6 − 92610000 x4 − 104186250 x2

C∗4 (x) = 276480 x9 + 3951360x7 + 30870000 x5 + 104186250x3

β5 = 7864320 , γ5 = −9282994875

40

A5(x) = −1769472 x11 + 38320128 x9 − 502951680x7 + 3300620400 x5 − 6188663250 x3+

+9282994875x

B5(x) = 19464192 x10 − 344881152 x8 + 3520661760x6 − 16503102000x4 + 18565989750x2

C5(x) = 2162688 x11 − 49268736 x9 + 704132352x7 − 5501034000 x5 + 18565989750x3

β∗5 = 7864320 , γ∗5 = −9282994875

A∗5(x) = 1769472 x11 + 38320128 x9 + 502951680x7 + 3300620400 x5 + 6188663250x3+

+9282994875x

B∗5(x) = −19464192 x10 − 344881152x8 − 3520661760 x6 − 16503102000x4 − 18565989750x2

C∗5 (x) = 2162688 x11 + 49268736 x9 + 704132352x7 + 5501034000 x5 + 18565989750x3

β6 = 125829120 , γ6 = 4867383646125

A6(x) = −28835840 x13 + 927793152x11 − 20092520448x9 + 263714330880x7 − 1730625296400x5+

+3244922430750x3 − 4867383646125x

B6(x) = 374865920 x12 − 10205724672x10 + 180832684032x8 − 1846000316160x6+

+8653126482000x4 − 9734767292250x2

C6(x) = 34078720 x13 − 1133969408 x11 + 25833240576x9 − 369200063232 x7 + 2884375494000x5−

−9734767292250x3

β∗6 = 125829120 , γ∗6 = −4867383646125

A∗6(x) = 28835840 x13 + 927793152x11 + 20092520448x9 + 263714330880x7 + 1730625296400x5+

+3244922430750x3 + 4867383646125x

B∗6(x) = −374865920x12 − 10205724672x10 − 180832684032x8 − 1846000316160x6−

−8653126482000x4 − 9734767292250x2

C∗6 (x) = 34078720 x13 + 1133969408x11 + 25833240576x9 + 369200063232x7 + 2884375494000x5+

+9734767292250x3

β7 = 7046430720 , γ7 = −12338817542926875

A7(x) = −1635778560 x15 + 73098854400x13 − 2351955640320x11 + 50934539335680x9−

−668515828780800x7 + 4387135126374000x5 − 8225878361951250x3 + 12338817542926875x

B7(x) = 24536678400 x14 − 950285107200x12 + 25871512043520x10 − 458410854021120x8+

+4679610801465600x6 − 21935675631870000x4 + 24677635085853750x2

C7(x) = 1887436800 x15 − 86389555200x13 + 2874612449280x11 − 65487264860160x9+

+935922160293120x7 − 7311891877290000x5 + 24677635085853750x3

β∗7 = 7046430720 , γ∗7 = −12338817542926875

A∗7(x) = 1635778560 x15 + 73098854400x13 + 2351955640320x11 + 50934539335680x9+

+668515828780800x7 + 4387135126374000x5 + 8225878361951250x3 + 12338817542926875x

B∗7(x) = −24536678400x14 − 950285107200 x12 − 25871512043520x10 − 458410854021120x8−

−4679610801465600x6 − 21935675631870000x4 − 24677635085853750x2

C∗7 (x) = 1887436800 x15 + 86389555200x13 + 2874612449280x11 + 65487264860160x9+

41

+935922160293120x7 + 7311891877290000x5 + 24677635085853750x3

β8 = 450971566080 , γ8 = 47195977101695296875

A8(x) = −105696460800 x17 + 6256852992000x15 − 279603118080000x13 + 8996230324224000x11−

−194824612958976000x9 + 2557073045086560000x7 − 16780791858380550000x5+

+31463984734463531250 x3 − 47195977101695296875x

B8(x) = 1796839833600 x16 − 93852794880000x14 + 3634840535040000x12 − 98958533566464000x10+

+1753421516630784000x8 − 17899511315605920000x6 + 83903959291902750000x4−

−94391954203390593750x2

C8(x) = 119789322240 x17 − 7219445760000x15 + 330440048640000x13 − 10995392618496000x11+

+250488788090112000 x9 − 3579902263121184000x7 + 27967986430634250000x5−

−94391954203390593750x3

β∗8 = 450971566080 , γ∗8 = −47195977101695296875

A∗8(x) = 105696460800 x17 + 6256852992000x15 + 279603118080000x13 + 8996230324224000x11+

+194824612958976000 x9 + 2557073045086560000x7 + 16780791858380550000x5+

+31463984734463531250 x3 + 47195977101695296875x

B∗8(x) = −1796839833600x16 − 93852794880000x14 − 3634840535040000x12 − 98958533566464000x10−

−1753421516630784000x8 − 17899511315605920000x6 − 83903959291902750000x4−

−94391954203390593750x2

C∗8 (x) = 119789322240 x17 + 7219445760000x15 + 330440048640000x13 + 10995392618496000x11+

+250488788090112000 x9 + 3579902263121184000x7 + 27967986430634250000x5+

+94391954203390593750 x3

42

XIII. Integrals of the type∫

x2n+1Zν(αx)Zν(βx) dx , α 6= β∫x · J0(αx)J0(βx) dx =

αxJ1(αx)J0(βx)− βxJ0(αx)J1(βx)α2 − β2∫

x · I0(αx)I0(βx) dx =αxI1(αx)I0(βx)− βxI0(αx)I1(βx)

α2 − β2∫x · J1(αx)J1(βx) dx =

x

α2 − β2[βJ1(αx)J0(βx)− αJ0(αx)J1(βx)]∫

x · I1(αx)I1(βx) dx =x

α2 − β2[αI0(αx)I1(βx)− βI1(αx)I0(βx)]∫

x3 · J0(αx)J0(βx) dx =

=2x2

(α2 − β2)2[(α2 + β2)J0(αx)J0(βx) + 2αβJ1(αx)J1(βx)

]+

+[4x · α2 + β2

(α2 − β2)3− x3

α2 − β2

]· [βJ0(αx)J1(βx)− αJ1(αx)J0(βx)]∫

x3 · I0(αx)I0(βx) dx =

=−2x2

(α2 − β2)2[(α2 + β2)I0(αx)I0(βx)− 2αβI1(αx)I1(βx)

]−

−[4x · α2 + β2

(α2 − β2)3+

x3

α2 − β2

]· [βI0(αx)I1(βx)− αI1(αx)I0(βx)]∫

x3 · J1(αx)J1(βx) dx =

=2x2

(α2 − β2)2[2αβJ0(αx)J0(βx) + (α2 + β2)J1(αx)J1(βx)

]+

+8αβx

(α2 − β2)3· [βJ0(αx)J1(βx)− αJ1(αx)J0(βx)]−

− x3

α2 − β2 · [αJ0(αx)J1(βx)− βJ1(αx)J0(βx)]∫x3 · I1(αx)I1(βx) dx =

=2x2

(α2 − β2)2[2αβI0(αx)I0(βx)− (α2 + β2)I1(αx)I1(βx)

]+

+8αβx

(α2 − β2)3· [βI0(αx)I1(βx)− αI1(αx)I0(βx)]+

+x3

α2 − β2 · [αI0(αx)I1(βx)− βI1(αx)I0(βx)]

Let ∫xm J0(αx)J0(βx) dx =

= Pm(x)J0(αx)J0(βx) + Qm(x)J0(αx)J1(βx) + Rm(x)J1(αx)J0(βx) + Sm(x)J1(αx)J1(βx) ,∫xm J1(αx)J1(βx) dx =

= Tm(x)J0(αx)J0(βx) + Um(x)J0(αx)J1(βx) + Vm(x)J1(αx)J0(βx) + Wm(x)J1(αx)J1(βx) ,∫xm I0(αx)I0(βx) dx =

43

= P ∗m(x)I0(αx)I0(βx) + Q∗m(x)I0(αx)I1(βx) + R∗m(x)I1(αx)I0(βx) + S∗m(x)I1(αx)I1(βx)

and ∫xm I1(αx)I1(βx) dx =

= T ∗m(x)I0(αx)I0(βx) + U∗m(x)I0(αx)I1(βx) + V ∗m(x)I1(αx)I0(βx) + W ∗m(x)I1(αx)I1(βx) ,

then holds

P5(x) = 4α2 + β2

(α2 − β2)2x4 − 32

α4 + β4 + 4 α2β2

(α2 − β2)4x2

Q5(x) = − β

α2 − β2x5 + 16

β(2 α2 + β2

)(α2 − β2)3

x3 − 64β(α4 + β4 + 4 α2β2

)(α2 − β2)5

x

R5(x) =α

α2 − β2x5 +−16

α(α2 + 2 β2

)(α2 − β2)3

x3 + 64α(α4 + β4 + 4 α2β2

)(α2 − β2)5

x

S5(x) = 8α β

(α2 − β2)2x4 − 96

α β(α2 + β2

)(α2 − β2)4

x2

P ∗5 (x) = −4α2 + β2

(α2 − β2)2x4 − 32

α4 + β4 + 4 α2β2

(α2 − β2)4x2

Q∗5(x) = − β

α2 − β2x5 − 16

β(2 α2 + β2

)(α2 − β2)3

x3 − 64β(α4 + β4 + 4 α2β2

)(α2 − β2)5

R∗5(x) =α

α2 − β2x5 + 16

α(α2 + 2 β2

)(α2 − β2)3

x3 + 64α(α4 + β4 + 4 α2β2

)(α2 − β2)5

x

S∗5 (x) = 8α β

(α2 − β2)2x4 + 96

α β(α2 + β2

)(α2 − β2)4

x2

T5(x) = 8α β

(α2 − β2)2x4 − 96

α β(α2 + β2

)(α2 − β2)4

x2

U5(x) = − α

α2 − β2x5 + 8

α(α2 + 5 β2

)(α2 − β2)3

x3 − 192α β2

(α2 + β2

)(α2 − β2)5

x

V5(x) =β

α2 − β2x5 − 8

β(5 α2 + β2

)(α2 − β2)3

x3 + 192β α2

(α2 + β2

)(α2 − β2)5

x

W5(x) = 4α2 + β2

(α2 − β2)2x4 − 16

β4 + α4 + 10 α2β2

(α2 − β2)4x2

T ∗5 (x) = 8α β

(α2 − β2)2x4 + 96

α β(α2 + β2

)(α2 − β2)4

x2

U∗5 (x) =α

α2 − β2x5 + 8

α(α2 + 5 β2

)(α2 − β2)3

x3 + 192α β2

(α2 + β2

)(α2 − β2)5

x

V ∗5 (x) = − β

α2 − β2x5 − 8

β(5 α2 + β2

)(α2 − β2)3

x3 − 192β α2

(α2 + β2

)(α2 − β2)5

x

W ∗5 (x) = −4

α2 + β2

(α2 − β2)2x4 − 16

β4 + α4 + 10 α2β2

(α2 − β2)4x2

P7(x) = 6α2 + β2

(α2 − β2)2x6 − 48

3 α4 + 14 α2β2 + 3 β4

(α2 − β2)4x4 + 1152

α6 + β6 + 9 α4β2 + 9 α2β4

(α2 − β2)6x2

Q7(x) = − β

α2 − β2x7 + 12

β(7 α2 + 3 β2

)(α2 − β2)3

x5−

44

−192β(8 α4 + 19 α2β2 + 3 β4

)(α2 − β2)5

x3 + 2304β(α6 + β6 + 9 α4β2 + 9 α2β4

)(α2 − β2)7

x

R7(x) =α

α2 − β2x7 − 12

α(3 α2 + 7 β2

)(α2 − β2)3

x5+

+192α(3 α4 + 19 α2β2 + 8 β4

)(α2 − β2)5

x3 − 2304α(α6 + β6 + 9 α4β2 + 9 α2β4

)(α2 − β2)7

x

S7(x) = 12α β

(α2 − β2)2x6 − 480

α β(α2 + β2

)(α2 − β2)4

x4 + 384α β

(38 α2β2 + 11 β4 + 11 α4

)(α2 − β2)6

x2

P ∗7 (x) = −6α2 + β2

(α2 − β2)2x6 − 48

3 α4 + 14 α2β2 + 3 β4

(α2 − β2)4x4 − 1152

β6 + 9 α4β2 + α6 + 9 α2β4

(α2 − β2)6x2

Q∗7(x) = − β

α2 − β2x7 − 12

β(7 α2 + 3 β2

)(α2 − β2)3

x5−

−192β(8 α4 + 19 α2β2 + 3 β4

)(α2 − β2)5

x3 − 2304β(β6 + 9 α4β2 + α6 + 9 α2β4

)(α2 − β2)7

x

R∗7(x) =α

α2 − β2x7 + 12

α(3 α2 + 7 β2

)(α2 − β2)3

x5+

+192α(3 α4 + 19 α2β2 + 8 β4

)(α2 − β2)5

x3 + 2304α(β6 + 9 α4β2 + α6 + 9 α2β4

)(α2 − β2)7

x

S∗7 (x) = 12α β

(α2 − β2)2x6 + 480

α β(α2 + β2

)(α2 − β2)4

x4 + 384α β

(11 α4 + 38 α2β2 + 11 β4

)(α2 − β2)6

x2

T7(x) = 12α β

(α2 − β2)2x6 − 480

α β(α2 + β2

)(α2 − β2)4

x4 + 4608α β

(α4 + β4 + 3 α2β2

)(α2 − β2)6

x2

U7(x) = − α

α2 − β2x7 + 24

α(α2 + 4 β2

)(α2 − β2)3

x5−

−192α(α4 + 18 α2β2 + 11 β4

)(α2 − β2)5

x3 + 9216α β2

(α4 + β4 + 3 α2β2

)(α2 − β2)7

x

V7(x) =β

α2 − β2x7 − 24

β(4 α2 + β2

)(α2 − β2)3

x5+

+192β(11 α4 + 18 α2β2 + β4

)(α2 − β2)5

x3 − 9216β α2

(α4 + β4 + 3 α2β2

)(α2 − β2)7

x

W7(x) = 6α2 + β2

(α2 − β2)2x6 − 96

α4 + 8 α2β2 + β4

(α2 − β2)4x4 + 384

β6 + 29 α4β2 + 29 α2β4 + α6

(α2 − β2)6x2

T ∗7 (x) = 12α β

(α2 − β2)2x6 + 480

α β(α2 + β2

)(α2 − β2)4

x4 + 4608α β

(β4 + 3 α2β2 + α4

)(α2 − β2)6

x2

U∗7 (x) =α

α2 − β2x7 + 24

α(α2 + 4 β2

)(α2 − β2)3

x5+

+192α(α4 + 18 α2β2 + 11 β4

)(α2 − β2)5

x3 + 9216α β2

(β4 + 3 α2β2 + α4

)(α2 − β2)7

x

V ∗7 (x) = − β

α2 − β2x7 − 24

β(4 α2 + β2

)(α2 − β2)3

x5−

−192β(11 α4 + 18 α2β2 + β4

)(α2 − β2)5

x3 − 9216β α2

(β4 + 3 α2β2 + α4

)(α2 − β2)7

x

45

W ∗7 (x) = −6

α2 + β2

(α2 − β2)2x6 − 96

α4 + 8 α2β2 + β4

(α2 − β2)4x4 − 384

α6 + 29 α4β2 + 29 α2β4 + β6

(α2 − β2)6x2

P9(x) = 8α2 + β2

(α2 − β2)2x8 − 384

α4 + 5 α2β2 + β4

(α2 − β2)4x6 + 3072

3 α6 + 32 α4β2 + 32 α2β4 + 3 β6

(α2 − β2)6x4−

−73728β8 + α8 + 16 α2β6 + 16 α6β2 + 36 α4β4

(α2 − β2)8x2

Q9(x) = − β

α2 − β2x9 + 32

β(5 α2 + 2 β2

)(α2 − β2)3

x7 − 768β(10 α4 + 22 α2β2 + 3 β4

)(α2 − β2)5

x5+

+6144β(19 α6 + 108 α4β2 + 77 α2β4 + 6 β6

)(α2 − β2)7

x3 − 147456β(β8 + α8 + 16 α2β6 + 16 α6β2 + 36 α4β4

)(α2 − β2)9

x

R9(x) =α

α2 − β2x9 − 32

α(2 α2 + 5 β2

)(α2 − β2)3

x7 + 768α(3 α4 + 22 α2β2 + 10 β4

)(α2 − β2)5

x5−

−6144α(6 α6 + 77 α4β2 + 108 α2β4 + 19 β6

)(α2 − β2)7

x3 + 147456α(β8 + α8 + 16 α2β6 + 16 α6β2 + 36 α4β4

)(α2 − β2)9

x

S9(x) = 16β α

(α2 − β2)2x8 − 1344

β α(α2 + β2

)(α2 − β2)4

x6 + 3072β α

(13 α4 + 44 α2β2 + 13 β4

)(α2 − β2)6

x4−

−61440β α

(5 β6 + 5 α6 + 37 α2β4 + 37 α4β2

)(α2 − β2)8

x2

P ∗9 (x) = −8α2 + β2

(α2 − β2)2x8 − 384

α4 + 5 α2β2 + β4

(α2 − β2)4x6 − 3072

3 α6 + 32 α4β2 + 32 α2β4 + 3 β6

(α2 − β2)6x4−

−73728α8 + 36 α4β4 + 16 α2β6 + 16 α6β2 + β8

(α2 − β2)8x2

Q∗9(x) = − β

α2 − β2x9 − 32

β(5 α2 + 2 β2

)(α2 − β2)3

x7 − 768β(10 α4 + 22 α2β2 + 3 β4

)(α2 − β2)5

x5−

−6144β(19 α6 + 108 α4β2 + 77 α2β4 + 6 β6

)(α2 − β2)7

x3 − 147456β(α8 + 36 α4β4 + 16 α2β6 + 16 α6β2 + β8

)(α2 − β2)9

x

R∗9(x) =α

α2 − β2x9 + 32

α(2 α2 + 5 β2

)(α2 − β2)3

x7 + 768α(3 α4 + 22 α2β2 + 10 β4

)(α2 − β2)5

x5+

+6144α(6 α6 + 77 α4β2 + 108 α2β4 + 19 β6

)(α2 − β2)7

x3 + 147456α(α8 + 36 α4β4 + 16 α2β6 + 16 α6β2 + β8

)(α2 − β2)9

x

S∗9 (x) = 16β α

(α2 − β2)2x8 + 1344

β α(α2 + β2

)(α2 − β2)4

x6 + 3072β α

(13 α4 + 44 α2β2 + 13 β4

)(α2 − β2)6

x4+

+61440β α

(5 β6 + 5 α6 + 37 α4β2 + 37 α2β4

)(α2 − β2)8

x2

T9(x) = 16β α

(α2 − β2)2x8 − 1344

β α(α2 + β2

)(α2 − β2)4

x6 + 1536β α

(27 α4 + 86 α2β2 + 27 β4

)(α2 − β2)6

x4−

−368640β α

(β6 + α6 + 6 α2β4 + 6 α4β2

)(α2 − β2)8

x2

U9(x) = − α

α2 − β2x9 + 16

α(3 α2 + 11 β2

)(α2 − β2)3

x7 − 384α(3 α4 + 43 α2β2 + 24 β4

)(α2 − β2)5

x5+

46

+3072α(3 α6 + 121 α4β2 + 239 α2β4 + 57 β6

)(α2 − β2)7

x3 − 737280α β2

(β6 + α6 + 6 α2β4 + 6 α4β2

)(α2 − β2)9

x

V9(x) =β

α2 − β2x9 − 16

β(11 α2 + 3 β2

)(α2 − β2)3

x7 + 384β(24 α4 + 43 α2β2 + 3 β4

)(α2 − β2)5

x5−

−3072β(57 α6 + 239 α4β2 + 121 α2β4 + 3 β6

)(α2 − β2)7

x3 + 737280β α2

(β6 + α6 + 6 α2β4 + 6 α4β2

)(α2 − β2)9

x

W9(x) = 8α2 + β2

(α2 − β2)2x8 − 96

3 α4 + 22 α2β2 + 3 β4

(α2 − β2)4x6 + 1536

3 α6 + 67 α4β2 + 67 α2β4 + 3 β6

(α2 − β2)6x4−

−61443 β8 + 3 α8 + 178 α2β6 + 178 α6β2 + 478 α4β4

(α2 − β2)8x2

T ∗9 (x) = 16β α

(α2 − β2)2x8 + 1344

β α(α2 + β2

)(α2 − β2)4

x6 + 1536β α

(27 α4 + 86 α2β2 + 27 β4

)(α2 − β2)6

x4+

+368640β α

(β6 + α6 + 6 α4β2 + 6 α2β4

)(α2 − β2)8

x2

U∗9 (x) =α

α2 − β2x9 + 16

α(3 α2 + 11 β2

)(α2 − β2)3

x7 + 384α(3 α4 + 43 α2β2 + 24 β4

)(α2 − β2)5

x5+

+3072α(3 α6 + 121 α4β2 + 239 α2β4 + 57 β6

)(α2 − β2)7

x3 + 737280α β2

(β6 + α6 + 6 α4β2 + 6 α2β4

)(α2 − β2)9

x

V ∗9 (x) = − β

α2 − β2x9 − 16

β(11 α2 + 3 β2

)(α2 − β2)3

x7 − 384β(24 α4 + 43 α2β2 + 3 β4

)(α2 − β2)5

x5−

−3072β(57 α6 + 239 α4β2 + 121 α2β4 + 3 β6

)(α2 − β2)7

x3 − 737280β α2

(β6 + α6 + 6 α4β2 + 6 α2β4

)(α2 − β2)9

x

W ∗9 (x) = −8

α2 + β2

(α2 − β2)2x8 − 96

3 α4 + 22 α2β2 + 3 β4

(α2 − β2)4x6 − 1536

3 α6 + 67 α4β2 + 67 α2β4 + 3 β6

(α2 − β2)6x4−

−61443 α8 + 478 α4β4 + 178 α2β6 + 178 α6β2 + 3 β8

(α2 − β2)8x2

P11 = 10α2 + β2

(α2 − β2)2x10 − 160

5 α4 + 26 α2β2 + 5 β4

(α2 − β2)4x8 + 7680

5 α6 + 58 α4β2 + 58 α2β4 + 5 β6

(α2 − β2)6x6−

−6144015 α8 + 283 α6β2 + 664 α4β4 + 283 α2β6 + 15 β8

(α2 − β2)8x4+

+7372800β10 + 100 β4α6 + 25 β2α8 + 25 β8α2 + α10 + 100 β6α4

(α2 − β2)10x2

Q11 = − β

α2 − β2x11 + 20

β(13 α2 + 5 β2

)(α2 − β2)3

x9 − 640β(37 α4 + 79 α2β2 + 10 β4

)(α2 − β2)5

x7+

+15360β(62 α6 + 332 α4β2 + 221 α2β4 + 15 β6

)(α2 − β2)7

x5−

−122880β(107 α8 + 1119 α6β2 + 1881 α4β4 + 643 α2β6 + 30 β8

)(α2 − β2)9

x3+

+14745600β(β10 + 100 β4α6 + 25 β2α8 + 25 β8α2 + α10 + 100 β6α4

)(α2 − β2)11

x

47

R11 =α

α2 − β2x11 − 20

α(5 α2 + 13 β2

)(α2 − β2)3

x9 + 640α(10 α4 + 79 α2β2 + 37 β4

)(α2 − β2)5

x7−

−15360α(15 α6 + 221 α4β2 + 332 α2β4 + 62 β6

)(α2 − β2)7

x5+

+122880α(30 α8 + 643 α6β2 + 1881 α4β4 + 1119 α2β6 + 107 β8

)(α2 − β2)9

x3−

−14745600α(β10 + 100 β4α6 + 25 β2α8 + 25 β8α2 + α10 + 100 β6α4

)(α2 − β2)11

x

S11 = 20α β

(α2 − β2)2x10 − 2880

α β(α2 + β2

)(α2 − β2)4

x8 + 3840α β

(47 α4 + 158 α2β2 + 47 β4

)(α2 − β2)6

x6−

−430080α β

(11 α6 + 79 α4β2 + 79 α2β4 + 11 β6

)(α2 − β2)8

x4+

+245760α β

(137 α8 + 1762 α6β2 + 3762 α4β4 + 1762 α2β6 + 137 β8

)(α2 − β2)10

x

P ∗11(x) = −10α2 + β2

(α2 − β2)2x10 − 160

5 α4 + 26 α2β2 + 5 β4

(α2 − β2)4x8 − 7680

5 α6 + 58 α4β2 + 58 α2β4 + 5 β6

(α2 − β2)6x6−

−6144015 α8 + 283 α6β2 + 664 α4β4 + 283 α2β6 + 15 β8

(α2 − β2)8x4−

−7372800α10 + β10 + 25 α8β2 + 100 α6β4 + 25 α2β8 + 100 α4β6

(α2 − β2)10x2

Q∗11(x) = − β

α2 − β2x11 − 20

β(13 α2 + 5 β2

)(α2 − β2)3

x9 − 640β(37 α4 + 79 α2β2 + 10 β4

)(α2 − β2)5

x7−

−15360β(62 α6 + 332 α4β2 + 221 α2β4 + 15 β6

)(α2 − β2)7

x5−

−122880β(107 α8 + 1119 α6β2 + 1881 α4β4 + 643 α2β6 + 30 β8

)(α2 − β2)9

x3−

−14745600β(α10 + β10 + 25 α8β2 + 100 α6β4 + 25 α2β8 + 100 α4β6

)(α2 − β2)11

x

R∗11(x) =α

α2 − β2x11 + 20

α(5 α2 + 13 β2

)(α2 − β2)3

x9 + 640α(10 α4 + 79 α2β2 + 37 β4

)(α2 − β2)5

x7+

+15360α(15 α6 + 221 α4β2 + 332 α2β4 + 62 β6

)(α2 − β2)7

x5+

+122880α(30 α8 + 643 α6β2 + 1881 α4β4 + 1119 α2β6 + 107 β8

)(α2 − β2)9

x3+

+14745600α(α10 + β10 + 25 α8β2 + 100 α6β4 + 25 α2β8 + 100 α4β6

)(α2 − β2)11

x

S∗11(x) = 20α β

(α2 − β2)2x10 + 2880

α β(α2 + β2

)(α2 − β2)4

x8 + 3840α β

(47 α4 + 158 α2β2 + 47 β4

)(α2 − β2)6

x6+

+430080α β

(11 α6 + 79 α4β2 + 79 α2β4 + 11 β6

)(α2 − β2)8

x4+

+245760α β

(137 α8 + 3762 α4β4 + 1762 α2β6 + 1762 α6β2 + 137 β8

)(α2 − β2)10

x2

48

T11 = 20α β

(α2 − β2)2x10 − 2880

α β(α2 + β2

)(α2 − β2)4

x8 + 46080α β

(4 α4 + 13 α2β2 + 4 β4

)(α2 − β2)6

x6−

−2580480α β

(2 α6 + 13 α4β2 + 13 α2β4 + 2 β6

)(α2 − β2)8

x4+

+44236800α β

(α8 + 10 α6β2 + 20 α4β4 + 10 α2β6 + β8

)(α2 − β2)10

x2

U11 = − α

α2 − β2x11 + 40

α(2 α2 + 7 β2

)(α2 − β2)3

x9 − 3840α(α4 + 13 α2β2 + 7 β4

)(α2 − β2)5

x7+

+92160α(α6 + 32 α4β2 + 59 α2β4 + 13 β6

)(α2 − β2)7

x5−

−737280α(α8 + 73 α6β2 + 300 α4β4 + 227 α2β6 + 29 β8

)(α2 − β2)9

x3+

+88473600α β2

(α8 + 10 α6β2 + 20 α4β4 + 10 α2β6 + β8

)(α2 − β2)11

x

V11 =β

α2 − β2x11 − 40

β(7 α2 + 2 β2

)(α2 − β2)3

x9 + 3840β(7 α4 + 13 α2β2 + β4

)(α2 − β2)5

x7−

−92160β(13 α6 + 59 α4β2 + 32 α2β4 + β6

)(α2 − β2)7

x5+

+737280β(29 α8 + 227 α6β2 + 300 α4β4 + 73 α2β6 + β8

)(α2 − β2)9

x3−

−88473600β α2

(α8 + 10 α6β2 + 20 α4β4 + 10 α2β6 + β8

)(α2 − β2)11

x

W11 = 10α2 + β2

(α2 − β2)2x10 − 640

α4 + 7 α2β2 + β4

(α2 − β2)4x8 + 23040

α6 + 20 α4β2 + 20 α2β4 + β6

(α2 − β2)6x6−

−368640α8 + 45 α6β2 + 118 α4β4 + 45 α2β6 + β8

(α2 − β2)8x4+

+1474560β10 + 527 β4α6 + 102 β2α8 + 102 β8α2 + α10 + 527 β6α4

(α2 − β2)10x2

T ∗11(x) = 20α β

(α2 − β2)2x10 + 2880

α β(α2 + β2

)(α2 − β2)4

x8 + 46080α β

(4 α4 + 13 α2β2 + 4 β4

)(α2 − β2)6

x6+

+2580480α β

(2 α6 + 13 α4β2 + 13 α2β4 + 2 β6

)(α2 − β2)8

x4+

+44236800α β

(α8 + 20 α4β4 + 10 α2β6 + 10 α6β2 + β8

)(α2 − β2)10

x2

U∗11(x) =α

α2 − β2x11 + 40

α(2 α2 + 7 β2

)(α2 − β2)3

x9 + 3840α(α4 + 13 α2β2 + 7 β4

)(α2 − β2)5

x7+

+92160α(α6 + 32 α4β2 + 59 α2β4 + 13 β6

)(α2 − β2)7

x5+

+737280α(α8 + 73 α6β2 + 300 α4β4 + 227 α2β6 + 29 β8

)(α2 − β2)9

x3+

+88473600α β2

(α8 + 20 α4β4 + 10 α2β6 + 10 α6β2 + β8

)(α2 − β2)11

x

49

V ∗11(x) = − β

α2 − β2x11 − 40

β(7 α2 + 2 β2

)(α2 − β2)3

x9 − 3840β(7 α4 + 13 α2β2 + β4

)(α2 − β2)5

x7−

−92160β(13 α6 + 59 α4β2 + 32 α2β4 + β6

)(α2 − β2)7

x5−

−737280β(29 α8 + 227 α6β2 + 300 α4β4 + 73 α2β6 + β8

)(α2 − β2)9

x3−

−88473600β α2

(α8 + 20 α4β4 + 10 α2β6 + 10 α6β2 + β8

)(α2 − β2)11

x

W ∗11(x) = −10

α2 + β2

(α2 − β2)2x10 − 640

α4 + 7 α2β2 + β4

(α2 − β2)4x8 − 23040

α6 + 20 α4β2 + 20 α2β4 + β6

(α2 − β2)6x6−

−368640α8 + 45 α6β2 + 118 α4β4 + 45 α2β6 + β8

(α2 − β2)8x4−

−1474560α10 + β10 + 102 α8β2 + 527 α6β4 + 102 α2β8 + 527 α4β6

(α2 − β2)10x2

Recurrence formulas: see page 54.

50

XIV. Integrals of the type∫

x2nZ0(αx)Z1(βx) dx , α 6= β∫x2 · J0(αx)J1(βx) dx =

=2βx

(α2 − β2)2[βJ0(αx)J1(βx)− αJ1(αx)J0(βx)] +

x2

α2 − β2 [βJ0(αx)J0(βx) + αJ1(αx)J1(βx)]∫x2 · I0(αx)I1(βx) dx =

= − 2βx

(α2 − β2)2[βI0(αx)I1(βx)− αI1(αx)I0(βx)]− x2

α2 − β2 [βI0(αx)I0(βx)− αI1(αx)I1(βx)]

Let ∫xm J0(αx)J1(βx) dx =

= Pm(x)J0(αx)J0(βx) + Qm(x)J0(αx)J1(βx) + Rm(x)J1(αx)J0(βx) + Sm(x)J1(αx)J1(βx)

and ∫xm I0(αx)I1(βx) dx =

= P ∗m(x)I0(αx)I0(βx) + Q∗m(x)I0(αx)I1(βx) + R∗m(x)I1(αx)I0(βx) + S∗m(x)I1(αx)I1(βx) ,

then holds

P4(x) =βx4

α2 − β2−

8β(2 α2 + β2

)x2

(α2 − β2)3

Q4(x) =2 (α2 + 2 β2)x3

(α2 − β2)2−

16β2(2 α2 + β2

)x

(α2 − β2)4

R4(x) =16 β α

(2 α2 + β2

)x

(α2 − β2)4− 6 β αx3

(α2 − β2)2

S4(x) =αx4

α2 − β2−

4 α(α2 + 5 β2

)x2

(α2 − β2)3

P ∗4 (x) = − β

α2 − β2x4 − 8

β(2 α2 + β2

)(α2 − β2)3

x2

Q∗4(x) = −2α2 + 2 β2

(α2 − β2)2x3 − 16

β2(2 α2 + β2

)(α2 − β2)4

x

R∗4(x) = 6α β

(α2 − β2)2x3 + 16

α β(2 α2 + β2

)(α2 − β2)4

x

S∗4 (x) =α

α2 − β2x4 + 4

α(α2 + 5 β2

)(α2 − β2)3

x2

P6(x) =β

α2 − β2x6 − 8

β(7 α2 + 3 β2

)(α2 − β2)3

x4 + 192β(3 α4 + β4 + 6 α2β2

)(α2 − β2)5

x2

Q6(x) = 22 α2 + 3 β2

(α2 − β2)2x5 − 32

α4 + 11 α2β2 + 3 β4

(α2 − β2)4x3 + 384

β2(3 α4 + β4 + 6 α2β2

)(α2 − β2)6

x

R6(x) = −10β α

(α2 − β2)2x5 + 32

β α(8 α2 + 7 β2

)(α2 − β2)4

x3 − 384β α

(3 α4 + β4 + 6 α2β2

)(α2 − β2)6

x

S6(x) =α

α2 − β2x6 − 16

α(α2 + 4 β2

)(α2 − β2)3

x4 + 64α(19 α2β2 + 10 β4 + α4

)(α2 − β2)5

x2

51

P ∗6 (x) = − β

α2 − β2x6 − 8

β(7 α2 + 3 β2

)(α2 − β2)3

x4 − 192β(6 α2β2 + 3 α4 + β4

)(α2 − β2)5

x2

Q∗6(x) = −22 α2 + 3 β2

(α2 − β2)2x5 − 32

α4 + 11 α2β2 + 3 β4

(α2 − β2)4x3 − 384

β2(6 α2β2 + 3 α4 + β4

)(α2 − β2)6

x

R∗6(x) = 10α β

(α2 − β2)2x5 + 32

α β(8 α2 + 7 β2

)(α2 − β2)4

x3 + 384α β

(6 α2β2 + 3 α4 + β4

)(α2 − β2)6

x

S∗6 (x) =α

α2 − β2x6 + 16

α(α2 + 4 β2

)(α2 − β2)3

x4 + 64α(10 β4 + 19 α2β2 + α4

)(α2 − β2)5

x2

P8(x) =β

α2 − β2x8 − 24

β(5 α2 + 2 β2

)(α2 − β2)3

x6 + 192β(21 α4 + 43 α2β2 + 6 β4

)(α2 − β2)5

x4−

−9216β(β6 + 4 α6 + 12 α2β4 + 18 α4β2

)(α2 − β2)7

x2

Q8(x) = 23 α2 + 4 β2

(α2 − β2)2x7 − 48

3 α4 + 26 α2β2 + 6 β4

(α2 − β2)4x5 + 384

3 α6 + 86 α4β2 + 109 α2β4 + 12 β6

(α2 − β2)6x3−

−18432β2(β6 + 4 α6 + 12 α2β4 + 18 α4β2

)(α2 − β2)8

x

R8(x) = −14β α

(α2 − β2)2x7 + 48

β α(18 α2 + 17 β2

)(α2 − β2)4

x5 − 1920β α

(9 α4 + 26 α2β2 + 7 β4

)(α2 − β2)6

x3+

+18432β α

(β6 + 4 α6 + 12 α2β4 + 18 α4β2

)(α2 − β2)8

x

S8(x) =α

α2 − β2x8 − 12

α(3 α2 + 11 β2

)(α2 − β2)3

x6 + 192α(3 α4 + 44 α2β2 + 23 β4

)(α2 − β2)5

x4−

−768α(47 β6 + 3 α6 + 239 α2β4 + 131 α4β2

)(α2 − β2)7

x2

P ∗8 (x) = − β

α2 − β2x8 − 24

β(5 α2 + 2 β2

)(α2 − β2)3

x6 − 192β(21 α4 + 43 α2β2 + 6 β4

)(α2 − β2)5

x4−

−9216β(β6 + 4 α6 + 18 α4β2 + 12 α2β4

)(α2 − β2)7

x2

Q∗8(x) = −23 α2 + 4 β2

(α2 − β2)2x7 − 48

3 α4 + 26 α2β2 + 6 β4

(α2 − β2)4x5 − 384

3 α6 + 86 α4β2 + 109 α2β4 + 12 β6

(α2 − β2)6x3−

−18432β2(β6 + 4 α6 + 18 α4β2 + 12 α2β4

)(α2 − β2)8

x

R∗8(x) = 14β α

(α2 − β2)2x7 + 48

β α(18 α2 + 17 β2

)(α2 − β2)4

x5 + 1920β α

(9 α4 + 26 α2β2 + 7 β4

)(α2 − β2)6

x3+

+18432β α

(β6 + 4 α6 + 18 α4β2 + 12 α2β4

)(α2 − β2)8

x

S∗8 (x) =α

α2 − β2x8 + 12

α(3 α2 + 11 β2

)(α2 − β2)3

x6 + 192α(3 α4 + 44 α2β2 + 23 β4

)(α2 − β2)5

x4+

+768α(47 β6 + 3 α6 + 131 α4β2 + 239 α2β4

)(α2 − β2)7

x2

52

P10 =β

α2 − β2x10 − 16

β(13 α2 + 5 β2

)(α2 − β2)3

x8 + 768β(19 α4 + 39 α2β2 + 5 β4

)(α2 − β2)5

x6−

−6144β(69 α6 + 332 α4β2 + 214 α2β4 + 15 β6

)(α2 − β2)7

x4+

+737280β(5 α8 + 40 α6β2 + 60 α4β4 + 20 α2β6 + β8

)(α2 − β2)9

x2

Q10 = 24 α2 + 5 β2

(α2 − β2)2x9 − 64

6 α4 + 47 α2β2 + 10 β4

(α2 − β2)4x7 + 1536

6 α6 + 136 α4β2 + 158 α2β4 + 15 β6

(α2 − β2)6x5−

−122886 α8 + 337 α6β2 + 1018 α4β4 + 499 α2β6 + 30 β8

(α2 − β2)8x3+

+1474560β2(5 α8 + 40 α6β2 + 60 α4β4 + 20 α2β6 + β8

)(α2 − β2)10

x

R10 = −18β α

(α2 − β2)2x9 + 64

β α(32 α2 + 31 β2

)(α2 − β2)4

x7 − 10752β α

(9 α4 + 28 α2β2 + 8 β4

)(α2 − β2)6

x5+

+12288β α

(144 α6 + 863 α4β2 + 782 α2β4 + 101 β6

)(α2 − β2)8

x3−

−1474560β α

(5 α8 + 40 α6β2 + 60 α4β4 + 20 α2β6 + β8

)(α2 − β2)10

x

S10 =α

α2 − β2x10 − 32

α(2 α2 + 7 β2

)(α2 − β2)3

x8 + 384α(6 α4 + 79 α2β2 + 41 β4

)(α2 − β2)5

x6−

−6144α(6 α6 + 199 α4β2 + 354 α2β4 + 71 β6

)(α2 − β2)7

x4+

+24576α(6 α8 + 481 α6β2 + 1881 α4β4 + 1281 α2β6 + 131 β8

)(α2 − β2)9

x2

P ∗10(x) = − β

α2 − β2x10 − 16

β(13 α2 + 5 β2

)(α2 − β2)3

x8 − 768β(19 α4 + 39 α2β2 + 5 β4

)(α2 − β2)5

x6−

−6144β(69 α6 + 332 α4β2 + 214 α2β4 + 15 β6

)(α2 − β2)7

x4−

−737280β(5 α8 + 60 α4β4 + 20 α2β6 + 40 α6β2 + β8

)(α2 − β2)9

x2

Q∗10(x) = −24 α2 + 5 β2

(α2 − β2)2x9−64

6 α4 + 47 α2β2 + 10 β4

(α2 − β2)4x7−1536

6 α6 + 136 α4β2 + 158 α2β4 + 15 β6

(α2 − β2)6x5−

−122886 α8 + 337 α6β2 + 1018 α4β4 + 499 α2β6 + 30 β8

(α2 − β2)8x3−

−1474560β2(5 α8 + 60 α4β4 + 20 α2β6 + 40 α6β2 + β8

)(α2 − β2)10

x

R∗10(x) = 18α β

(α2 − β2)2x9 + 64

α β(32 α2 + 31 β2

)(α2 − β2)4

x7 + 10752α β

(9 α4 + 28 α2β2 + 8 β4

)(α2 − β2)6

x5+

+12288α β

(144 α6 + 863 α4β2 + 782 α2β4 + 101 β6

)(α2 − β2)8

x3+

53

+1474560α β

(5 α8 + 60 α4β4 + 20 α2β6 + 40 α6β2 + β8

)(α2 − β2)10

x

S∗10(x) =α

α2 − β2x9 + 32

α(2 α2 + 7 β2

)(α2 − β2)3

x7 + 384α(6 α4 + 79 α2β2 + 41 β4

)(α2 − β2)5

x5+

+6144α(6 α6 + 199 α4β2 + 354 α2β4 + 71 β6

)(α2 − β2)7

x3+

−24576α(6 α8 + 1881 α4β4 + 1281 α2β6 + 481 α6β2 + 131 β8

)(α2 − β2)9

x

Recurrence formulas:

LetZ(m)

µν =∫

xmJµ(αx)Jν(βx) dx and Z∗(m)µν =

∫xmIµ(αx)Iν(βx) dx ,

then the following system holds:

Z(2n+3)00 = x2n+2 · Z(1)

00 −2(n + 1)α2 − β2

[αZ

(2n+2)10 − βZ

(2n+2)01

]Z

(2n+3)11 = x2n+2 · Z(1)

11 −2(n + 1)α2 − β2

[βZ

(2n+2)10 − αZ

(2n+2)01

]Z

(2n+2)01 = x2n · Z(2)

01 −4nβ

(α2 − β2)2[βZ

(2n)01 − αZ

(2n)10

]− 2n

α2 − β2

[βZ

(2n+1)00 + αZ

(2n+1)11

]Z

(2n+2)10 = x2n · Z(2)

10 +4nα

(α2 − β2)2[βZ

(2n)01 + αZ

(2n)10

]+

2n

α2 − β2

[αZ

(2n+1)00 + βZ

(2n+1)11

]respectively

Z∗(2n+3)00 = x2n+2 · Z∗(1)00 − 2(n + 1)

α2 − β2

[αZ

∗(2n+2)10 − βZ

∗(2n+2)01

]Z∗(2n+3)11 = x2n+2 · Z∗(1)11 +

2(n + 1)α2 − β2

[βZ

∗(2n+2)10 − αZ

∗(2n+2)01

]Z∗(2n+2)01 = x2n · Z∗(2)01 +

4nβ

(α2 − β2)2[βZ

∗(2n)01 − αZ

∗(2n)10

]+

2n

α2 − β2

[βZ

∗(2n+1)00 − αZ

∗(2n+1)11

]Z∗(2n+2)10 = x2n · Z∗(2)10 − 4nα

(α2 − β2)2[βZ

∗(2n)01 − αZ

∗(2n)10

]− 2n

α2 − β2

[αZ

∗(2n+1)00 − βZ

∗(2n+1)11

]

54

XV. Integrals of the type∫

x2n · Zν(αx) · Zν(βx) dx

a) Basic Integrals:In the case α = β it was necessary to define the new functions Θ(x) and Ω(x) (see page 21). The morethere is no solution expected in the described class of functions if α 6= β.Let 0 < β < α and γ = β/α < 1. The integrals may be reduced to the single parameter γ by∫

x2n · Zν(αx) · Zν(βx) dx = α−2n−1

∫t2n · Zν(t) · Zν(γt) dt , t = αx .

The functions Θ(x) and Ω(x) from page 21 are generalized to

Θ0(x; γ) =∫ x

0

J0(s) · J0(γs) ds and Ω0(x; γ) =∫ x

0

I0(s) · I0(γs) ds .

Power series:

Θ0(x; γ) =∞∑

k=0

(−1)k

(k!)2 · 4k · (2k + 1)· (1− γ2)k · Pk

(1 + γ2

1− γ2

)x2k+1

and

Ω0(x; γ) =∞∑

k=0

(1− γ2)k

(k!)2 · 4k · (2k + 1)· Pk

(1 + γ2

1− γ2

)x2k+1 ,

wherePn (x) =

(2n)!2n · (n!)2

xn + . . .

denotes the Legendre polynoms. Their values may be found by the recurrence relation

Pn+1

(1 + γ2

1− γ2

)=

2n + 1n + 1

· 1 + γ2

1− γ2· Pn

(1 + γ2

1− γ2

)− n

n + 1Pn−1

(1 + γ2

1− γ2

)with

P0

(1 + γ2

1− γ2

)= 1 and P1

(1 + γ2

1− γ2

)=

1 + γ2

1− γ2.

Some first terms of the power series:

Θ0(x; γ) = x− γ2 + 112

x3 +γ4 + 4 γ2 + 1

320x5− γ6 + 9 γ4 + 9 γ2 + 1

16128x7 +

γ8 + 16 γ6 + 36 γ4 + 16 γ2 + 11327104

x9−

−γ10 + 25 γ8 + 100 γ6 + 100 γ4 + 25 γ2 + 1162201600

x11 +γ12 + 36 γ10 + 225 γ8 + 400 γ6 + 225 γ4 + 36 γ2 + 1

27603763200x13−

−γ14 + 49 γ12 + 441 γ10 + 1225 γ8 + 1225 γ6 + 441 γ4 + 49 γ2 + 16242697216000

x15+

+γ16 + 64 γ14 + 784 γ12 + 3136 γ10 + 4900 γ8 + 3136 γ6 + 784 γ4 + 64 γ2 + 1

1811214552268800x17−

−γ18 + 81 γ16 + 1296 γ14 + 7056 γ12 + 15876 γ10 + 15876 γ8 + 7056 γ6 + 1296 γ4 + 81 γ2 + 1655872751986278400

x19 + . . .

If x > γx >> 1 one has

Ω0(x; γ) ≈ e(1+γ)x

2π√

γ(1 + γ)x.

Let Θ0(x; γ) be computed with n decimal signs, then in the case x > γx >> 1 the loss of significant digitscan be expected. Only about

n− lge(1+γ)x

2π√

γ(1 + γ)x

significant digits are left.The upper integral with primary parameters:∫ x

0

J0(αs) · J0(βs) ds =∞∑

k=0

(−1)k

(k!)2 · 4k · (2k + 1)· (α2 − β2)k · Pk

(α2 + β2

α2 − β2

)x2k+1

55

Asymptotic series for x > γx >> 1:

Θ0(x; γ) ∼ 2π

K(γ) +1

π√

γ x

[sin(1− γ)x

1− γ− cos(γ + 1)x

γ + 1

]+

+1

8πγ3/2 x2

[γ2 − 10 γ + 1

(1− γ)2cos(1− γ)x− γ2 + 10 γ + 1

(γ + 1)2sin(γ + 1)x

]+

+1

128πγ5/2 x3

[9 γ4 + 52 γ3 + 342 γ2 + 52 γ + 9

(γ + 1)3cos(γ + 1)x− 9 γ4 − 52 γ3 + 342 γ2 − 52 γ + 9

(1− γ)3sin(1− γ)x

]+

+3

1024πγ7/2 x4

[25 γ6 + 150 γ5 + 503 γ4 + 2804 γ3 + 503 γ2 + 150 γ + 25

(γ + 1)4sin(γ + 1)x−

− 25 γ6 − 150 γ5 + 503 γ4 − 2804 γ3 + 503 γ2 − 150 γ + 25(1− γ)4

cos(1− γ)x]

+ . . . ,

where K denotes the complete elliptic integral of the first kind, see [1] or [5].Particulary follows

limx→∞

Θ0(x; γ) =2π

K(γ) .

Some values of this limit:

γ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 1.0000 1.0000 1.0001 1.0002 1.0004 1.0006 1.0009 1.0012 1.0016 1.00200.1 1.0025 1.0030 1.0036 1.0043 1.0050 1.0057 1.0065 1.0073 1.0083 1.00920.2 1.0102 1.0113 1.0124 1.0136 1.0149 1.0162 1.0176 1.0190 1.0205 1.02210.3 1.0237 1.0254 1.0272 1.0290 1.0309 1.0329 1.0350 1.0371 1.0394 1.04170.4 1.0441 1.0465 1.0491 1.0518 1.0545 1.0574 1.0603 1.0634 1.0665 1.06980.5 1.0732 1.0767 1.0803 1.0841 1.0880 1.0920 1.0962 1.1006 1.1051 1.10970.6 1.1146 1.1196 1.1248 1.1302 1.1359 1.1417 1.1479 1.1542 1.1609 1.16780.7 1.1750 1.1826 1.1905 1.1988 1.2074 1.2166 1.2262 1.2363 1.2470 1.25830.8 1.2702 1.2830 1.2965 1.3110 1.3265 1.3432 1.3613 1.3809 1.4023 1.42580.9 1.4518 1.4810 1.5139 1.5517 1.5959 1.6489 1.7145 1.8004 1.9232 2.1369

The following picture shows Θ0(x; 1/2) (solid line), the asymptotic approximation with x−1 (long dashes)and with x−2 (short dashes):

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................Θ0(x; γ)

...............0.2

...............0.4

...............0.6

...............0.8

...............1.0

...............1.2

...............1.4

1.0732

...........3 ...........6 ...........9 ...........12 ...........15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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....................................

...........

...........

...........

...........

...........

...........

...........

...........

........................................................................................................................................................................................... ........... ........... ............................................ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ...........

Figure 5 : Function Θ0 (x; γ) with γ = 0.5

56

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................Θ0(x; γ)

...............0.2

...............0.4

...............0.6

...............0.8

...............1.0

...............1.2

...............1.4

...........3

...........6

...........9

...........12

...........15

............................................................................................................................................... γ = 1/4

............................. ............................. ............................. γ = 1/3

............... ............... ............... ............... ............... . γ = 2/3

. . . . . . . . . . . . γ = 3/4

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...............................................................................................................................................................................................................................................................

............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... .............

......................................

. . . . . . . . . . . .. . .

. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 6 : Some Functions Θ0 (x; γ)Let

Θ1(x; γ) =∫ x

0

J1(s) · J1(γs) ds and Ω1(x; γ) =∫ x

0

I1(s) · I1(γs) ds .

Power series:

Θ1(x; γ) =∞∑

k=0

(−1)k

(k!)2 · 4k+1 · (k + 1) · (2k + 3)· (1− γ2)k+1 · P−1

k+1

(1 + γ2

1− γ2

)x2k+3

and

Ω1(x; γ) =∞∑

k=0

(1− γ2)k+1

(k!)2 · 4k+1 · (k + 1) · (2k + 3)· P−1

k+1

(1 + γ2

1− γ2

)x2k+3 ,

where P−1n (x) denotes the associated Legendre functions of the first kind. Their values may be found by

the recurrence relation, starting with n = 0:

P−1n+1

(1 + γ2

1− γ2

)=

2n + 1n + 2

· 1 + γ2

1− γ2· P−1

n

(1 + γ2

1− γ2

)− n− 1

n + 2P−1

n−1

(1 + γ2

1− γ2

)with

P−10

(1 + γ2

1− γ2

)= γ and P−1

1

(1 + γ2

1− γ2

)=

γ

1− γ2.

Some first terms of the power series:

Θ1(x; γ) =γ

12x3−γ3 + γ

160x5+

γ5 + 3 γ3 + γ

5376x7−γ7 + 6 γ5 + 6 γ3 + γ

331776x9+

γ9 + 10 γ7 + 20 γ5 + 10 γ3 + γ

32440320x11−

−γ11 + 15 γ9 + 50 γ7 + 50 γ5 + 15 γ3 + γ

4600627200x13 +

γ13 + 21 γ11 + 105 γ9 + 175 γ7 + 105 γ5 + 21 γ3 + γ

891813888000x15−

−γ15 + 28 γ13 + 196 γ11 + 490 γ9 + 490 γ7 + 196 γ5 + 28 γ3 + γ

226401819033600x17+

+γ17 + 36 γ15 + 336 γ13 + 1176 γ11 + 1764 γ9 + 1176 γ7 + 336 γ5 + 36 γ3 + γ

72874750220697600x19 − . . .

In the case x > γx >> 1 one has once again

Ω1(x; γ) ≈ e(1+γ)x

2π√

γ(1 + γ)x.

57

Asymptotic series for x > γx >> 1:

Θ1(x; γ) ∼ 2πγ

[K(γ)−E(γ) ] +1

π√

γ

[1x

(cos(1 + γ)x

1 + γ− sin(1− γ)x

1− γ

)−

− 18g x2

(3γ2 − 2γ + 3

(1 + γ)2sin(1 + γ)x +

3γ2 + 2γ + 3(1− γ)2

cos(1− γ)x)

+

+1

128g3 x3

(15γ4 + 108γ3 − 70γ2 + 108γ + 15

(1 + γ)3cos(1 + γ)x− 15γ4 − 108γ3 − 70γ2 − 108γ + 15

(1− γ)3sin(1− γ)x

)−

− 31024γ3 x4

(35γ6 + 210γ5 + 909γ4 − 580γ3 + 909γ2 + 210γ + 35

(1 + γ)4sin(1 + γ)x+

+35γ6 − 210γ5 + 909γ4 + 580γ3 + 909γ2 − 210γ + 35

(1− γ)4cos(1− γ)x

)+ . . .

]with the complete elliptic integrals of the first and second kind.Particulary follows

limx→∞

Θ1(x; γ) =2

πγ[K(γ)−E(γ)] .

Some values of this limit:

γ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0351 0.0401 0.04510.1 0.0502 0.0553 0.0603 0.0654 0.0705 0.0756 0.0808 0.0859 0.0911 0.09630.2 0.1015 0.1068 0.1121 0.1174 0.1227 0.1280 0.1334 0.1389 0.1443 0.14980.3 0.1554 0.1609 0.1666 0.1722 0.1780 0.1837 0.1895 0.1954 0.2013 0.20730.4 0.2134 0.2195 0.2257 0.2319 0.2382 0.2446 0.2511 0.2577 0.2643 0.27110.5 0.2779 0.2849 0.2919 0.2991 0.3064 0.3138 0.3214 0.3290 0.3369 0.34480.6 0.3530 0.3613 0.3698 0.3785 0.3873 0.3964 0.4058 0.4153 0.4252 0.43530.7 0.4457 0.4564 0.4674 0.4789 0.4907 0.5029 0.5157 0.5289 0.5427 0.55710.8 0.5721 0.5879 0.6046 0.6221 0.6407 0.6605 0.6816 0.7042 0.7287 0.75520.9 0.7844 0.8165 0.8525 0.8934 0.9407 0.9967 1.0654 1.1543 1.2803 1.4971

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..............................................

............................Θ1(x; γ)

...............0.2

...............0.4

...............0.6

................

......3......................

6......................9

................

......12......................15

........................................................................................................... γ = 1/4 ............................. ............................. γ = 1/3 ...................... ...................... ...................... γ = 1/2

............... ............... ............... ............... γ = 2/3

. . . . . . . . . γ = 3/4

...................... ..................................................................................................................................................................................................................................................

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............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ............... ....

. . . . ........................................ . . . . . . . . . . . . . .

. . .. . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 7 : Some Functions Θ1 (x; γ)

58

The value of x may be too large to use the power series for Θ0(x; γ) and γx may be too small to apply theasymptotic formula. In this case

Θ0(x; γ) ∼ 2π

K(γ)+

+A0(x; γ) cos xJ0(γx) + A1(x; γ) cos xJ1(γx) + B0(x; γ) sin xJ0(γx) + B1(x; γ) sin xJ1(γx)√

πx

is applicable. Let

Aµ(x; γ) =∞∑

k=0

a(µ)k (x; γ)

(1− γ2)k+1 xkand Bµ(x; γ) =

∞∑k=0

b(µ)k (x; γ)

(1− γ2)k+1 xk,

then holds

a(0)0 (x; γ) = −1 , a

(0)1 (x; γ) = −11γ2 + 5

8, a

(0)2 (x; γ) = −31 γ4 − 926 γ2 − 129

128,

a(0)3 (x; γ) =

3(γ2 + 15

) (59 γ4 + 906 γ2 + 59

)1024

,

a(0)4 (x; γ) =

7125 γ8 + 15468 γ6 − 4088898 γ4 − 8215572 γ2 − 30103532768

,

a(0)5 (x; γ) = −102165 γ10 − 208569 γ8 + 25390098 γ6 + 501398862 γ4 + 469053609 γ2 + 10896795

262144,

a(0)6 (x; γ) =

−45 [84231 γ12 − 348490 γ10 + 2847497 γ8 − 451498956 γ6 − 2481377623 γ4 − 1343311306 γ2 − 21362649]4194304

a(1)0 (x; γ) = −γ , a

(1)1 (x; γ) = −

γ(γ2 − 17

)8

, a(1)2 (x; γ) =

γ(9 γ4 + 206 γ2 + 809

)128

,

a(1)3 (x; γ) =

γ(75 γ6 + 143 γ4 − 24063 γ2 − 25307

)1024

,

a(1)4 (x; γ) = −

3γ(1225 γ8 − 1892 γ6 + 201078 γ4 + 2678812 γ2 + 1315081

)32768

,

a(1)5 (x; γ) = −

3γ(19845 γ10 − 67625 γ8 + 467314 γ6 − 58112658 γ4 − 216355367 γ2 − 61495829

)262144

,

a(1)6 (x; γ) =

3γ(800415 γ12 − 3869530 γ10 + 12921201 γ8 + 680167252 γ6 + 20763422609 γ4 + 36255061542 γ2 + 6716005951

)4194304

b(0)0 (x; γ) = 1 , b

(0)1 (x; γ) = −11γ2 + 5

8= a

(0)1 , b

(0)2 (x; γ) =

31 γ4 − 926 γ2 − 129128

= −a(0)2 ,

b(0)3 (x; γ) =

3(γ2 + 15

) (59 γ4 + 906 γ2 + 59

)1024

= a(0)3 ,

b(0)4 (x; γ) = −7125 γ8 + 15468 γ6 − 4088898 γ4 − 8215572 γ2 − 301035

32768= −a

(0)4 ,

b(0)5 (x; γ) = −102165 γ10 − 208569 γ8 + 25390098 γ6 + 501398862 γ4 + 469053609 γ2 + 10896795

262144= a

(0)5 ,

b(0)6 (x; γ) = −a

(0)6 =

45 [84231 γ12 − 348490 γ10 + 2847497 γ8 − 451498956 γ6 − 2481377623 γ4 − 1343311306 γ2 − 21362649]4194304

59

b(1)0 (x; γ) = −γ , b

(1)1 (x; γ) =

γ(γ2 − 17

)8

= a(1)1 , b

(1)2 (x; γ) =

γ(9 γ4 + 206 γ2 + 809

)128

= a(1)2 ,

b(1)3 (x; γ) = −

γ(75 γ6 + 143 γ4 − 24063 γ2 − 25307

)1024

= −a(0)3 ,

b(1)4 (x; γ) = −

3γ(1225 γ8 − 1892 γ6 + 201078 γ4 + 2678812 γ2 + 1315081

)32768

= a(1)4 ,

b(1)5 (x; γ) =

3γ(19845 γ10 − 67625 γ8 + 467314 γ6 − 58112658 γ4 − 216355367 γ2 − 61495829

)262144

= −a(1)5 ,

b(1)6 (x; γ) = a

(1)6 =

3γ(800415 γ12 − 3869530 γ10 + 12921201 γ8 + 680167252 γ6 + 20763422609 γ4 + 36255061542 γ2 + 6716005951

)4194304

When γ << 1 one has approximately

A0(x; γ) ≈ A0(x; 0) = −1− 0.625x

+1.0078

x2+

2.5928x3

− 9.1869x4

− 41.568x5

+229.20

x6+ . . .

B0(x; γ) ≈ B0(x; 0) = 1− 0.625x

− 1.0078x2

+2.5928

x3+

9.1869x4

− 41.568x5

− 229.20x6

+ . . .

A1(x; γ) ≈ γ∂A1

∂γ(x; 0) = γ

[−1 +

2.125x

+6.3203

x2− 24.714

x3− 120.40

x4+

703.76x5

+4803.7

x6+ . . .

]B1(x; γ) ≈ γ

∂B1

∂γ(x; 0) = γ

[−1− 2.125

x+

6.3203x2

+24.714

x3− 120.40

x4− 703.76

x5+

4803.7x6

+ . . .

]∣∣∣∣∣a(0)

3 (x; 0)

a(0)2 (x; 0)

∣∣∣∣∣ =∣∣∣∣∣b(0)

3 (x; 0)

b(0)2 (x; 0)

∣∣∣∣∣ = 2.57 ,

∣∣∣∣∣a(0)4 (x; 0)

a(0)3 (x; 0)

∣∣∣∣∣ = 3.54 ,

∣∣∣∣∣a(0)5 (x; 0)

a(0)4 (x; 0)

∣∣∣∣∣ = 4.52 ,

∣∣∣∣∣a(0)6 (x; 0)

a(0)5 (x; 0)

∣∣∣∣∣ = 5.51

The summand a(0)k (x; γ)/[(1− γ2)k+1 xk] can be used if |x| > |a(0)

k (x; γ)/a(0)k−1(x; γ)|.

The same holds for b(0)k (x; γ).

Let∆n(x; γ) = −Θ0(x; γ)+

+1√πx

[A

(n)0 (x; γ) cos xJ0(γx) + A

(n)1 (x; γ) cos xJ1(γx) + B

(n)0 (x; γ) sin xJ0(γx) + B

(n)1 (x; γ) sin xJ1(γx)

]with

A(n)µ (x; γ) =

n∑k=0

a(µ)k (x; γ)

(1− γ2)k+1 xkand B(n)

µ (x; γ) =n∑

k=0

b(µ)k (x; γ)

(1− γ2)k+1 xk.

For the case γ = 0.1 some of these differences are shown:

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x

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0.025

0.050

0.075

0.100

-0.025

-0.050

........

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.............4

.............6

.............8

.............10

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∆0(x, 0.1)

∆1(x, 0.1)

60

Figure 8 : Differences ∆0 (x; γ) and ∆1 (x; γ) with γ = 0.1

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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0.025

0.050

-0.025........

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...........6

...........8

...........10

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............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............

∆2(x, 0.1)

∆1(x, 0.1)


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x

........0.015

-0.02 ........

...............4

...............6

...............8

...............10

......................................................................................................................................................................................................................................................................

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........... ............. .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......

∆2(x, 0.1)

∆3(x, 0.1)


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x

........

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........0.001

0.002

-0.001

................

.............8

................

.............10

................

.............12

................

.............14

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............................................................................................................................................................................................................................................................................................................................................................................................................................................

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............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .........

∆4(x, 0.1)

∆3(x, 0.1)


The same way the asymptotic expansion

Θ1(x; γ) ∼ 2πγ

[K(γ)−E(γ)]+

+A∗0(x; γ) cos xJ0(γx) + A∗1(x; γ) cos xJ1(γx) + B∗0(x; γ) sin xJ0(γx) + B∗1(x; γ) sin xJ1(γx)√

πx

is applicable in the case x >> 1 and γx ≈ 1. Let

A∗µ(x; γ) =∞∑

k=0

a(µ,∗)k (x; γ)

(1− γ2)k+1 xkand B∗µ(x; γ) =

∞∑k=0

b(µ,∗)k (x; γ)

(1− γ2)k+1 xk,

61

then holds

a(0,∗)0 (x; γ) = −γ , a

(0,∗)1 (x; γ) = −

γ(3 γ2 + 13

)8

, a(0,∗)2 (x; γ) = −

γ(15 γ4 − 382 γ2 − 657

)128

,

a(0,∗)3 (x; γ) =

3 γ(35 γ6 + 327 γ4 + 8457 γ2 + 7565

)1024

a(0,∗)4 (x; γ) =

3 γ(1575 γ8 − 860 γ6 − 455382 γ4 − 2435292 γ2 − 1304345

)32768

a(0,∗)5 (x; γ) = −

3 γ(24255 γ10 − 74795 γ8 + 1750326 γ6 + 76252650 γ4 + 190548859 γ2 + 67043025

)262144

a(0,∗)6 (x; γ) =

45 γ(63063 γ12 − 295722 γ10 + 1295545 γ8 − 137114124 γ6 − 1469273511 γ4 − 2157119402 γ2 − 532523145

)4194304

a(1,∗)0 (x; γ) = −1 , a

(1,∗)1 (x; γ) =

7 γ2 + 98

, a(1,∗)2 (x; γ) =

57 γ4 + 622 γ2 + 345128

,

a(1,∗)3 (x; γ) =

195 γ6 − 8921 γ4 − 30871 γ2 − 95551024

a(1,∗)4 (x; γ) = −7035 γ8 + 100692 γ6 + 4097826 γ4 + 7006164 γ2 + 1371195

32768

a(1,∗)5 (x; γ) = −97335 γ10 − 38595 γ8 − 54339354 γ6 − 442588230 γ4 − 449504301 γ2 − 60259815

262144

a(1,∗)6 (x; γ) =

3565485 γ12 − 12841710 γ10 + 423532419 γ8 + 25838749116 γ6 + 96291507171 γ4 + 64464832914 γ2 + 62641829254194304

b(0,∗)0 (x; γ) = γ , b

(0,∗)1 (x; γ) = −

γ(3 γ2 + 13

)8

= a(0,∗)1 , b

(0,∗)2 (x; γ) =

γ(15 γ4 − 382 γ2 − 657

)128

= −a(0,∗)2 ,

b(0,∗)3 (x; γ) =

3 γ(35 γ6 + 327 γ4 + 8457 γ2 + 7565

)1024

= a(0,∗)3

b(0,∗)4 (x; γ) = −

3 γ(1575 γ8 − 860 γ6 − 455382 γ4 − 2435292 γ2 − 1304345

)32768

= −a(0,∗)4

b(0,∗)5 (x; γ) = −

3 γ(24255 γ10 − 74795 γ8 + 1750326 γ6 + 76252650 γ4 + 190548859 γ2 + 67043025

)262144

= a(0,∗)5

b(0,∗)6 (x; γ) = −a

(0,∗)6 =

45 γ(63063 γ12 − 295722 γ10 + 1295545 γ8 − 137114124 γ6 − 1469273511 γ4 − 2157119402 γ2 − 532523145

)4194304

b(1,∗)0 (x; γ) = −1 , b

(1,∗)1 (x; γ) = −7 γ2 + 9

8= −a

(1,∗)1 , b

(1,∗)2 (x; γ) =

57 γ4 + 622 γ2 + 345128

= a(1,∗)2 ,

b(1,∗)3 (x; γ) =

195 γ6 − 8921 γ4 − 30871 γ2 − 95551024

= −a(1,∗)3

b(1,∗)4 (x; γ) = −7035 γ8 + 100692 γ6 + 4097826 γ4 + 7006164 γ2 + 1371195

32768= a

(1,∗)4

b(1,∗)5 (x; γ) =

97335 γ10 − 38595 γ8 − 54339354 γ6 − 442588230 γ4 − 449504301 γ2 − 60259815262144

= −a(1,∗)5

b(1,∗)6 (x; γ) = a

(1,∗)6 =

62

3565485 γ12 − 12841710 γ10 + 423532419 γ8 + 25838749116 γ6 + 96291507171 γ4 + 64464832914 γ2 + 62641829254194304

When γ << 1 one has approximately

A∗0(x; γ) ≈ γ∂A∗0∂γ

(x; 0) = γ

[−1− 1.625

x+

5.1328x2

+22.163

x3− 119.42

x4− 767.25

x5+

5713.4x6

+ . . .

]

B∗0(x; γ) ≈ γ∂B∗0∂γ

(x; 0) = γ

[1− 1.625

x− 5.1328

x2+

22.163x3

+119.42

x4− 767.25

x5− 5713.4

x6+ . . .

]A∗1(x; γ) ≈ A∗1(x; 0) = −1 +

1.125x

+2.6953

x2− 9.3311

x3− 41.846

x4+

229.87x5

+1493.5

x6+ . . .

B∗1(x; γ) ≈ B∗1(x; 0) = −1− 1.125x

+2.6953

x2+

9.3311x3

− 41.846x4

− 229.87x5

+1493.5

x6+ . . .∣∣∣∣∣a(1,∗)

3 (x; 0)

a(1,∗)2 (x; 0)

∣∣∣∣∣ =∣∣∣∣∣b(1,∗)

3 (x; 0)

b(1,∗)2 (x; 0)

∣∣∣∣∣ = 3.46 ,

∣∣∣∣∣a(1,∗)4 (x; 0)

a(1,∗)3 (x; 0)

∣∣∣∣∣ = 4.48 ,

∣∣∣∣∣a(1,∗)5 (x; 0)

a(1,∗)4 (x; 0)

∣∣∣∣∣ = 5.49 ,

∣∣∣∣∣a(1,∗)6 (x; 0)

a(1,∗)5 (x; 0)

∣∣∣∣∣ = 6.50

The summand a(0,∗)k (x; γ)/[(1− γ2)k+1 xk] can be used if |x| > |a(0,∗)

k (x; γ)/a(0)k−1(x; γ)|.

The same holds for b(0,∗)k (x; γ).

Let∆∗

n(x; γ) = −Θ1(x; γ)+

+1√πx

[A

(n,∗)0 (x; γ) cos xJ0(γx) + A

(n,∗)1 (x; γ) cos xJ1(γx) + B

(n,∗)0 (x; γ) sinxJ0(γx) + B

(n,∗)1 (x; γ) sin xJ1(γx)

]with

A(n,∗)µ (x; γ) =

n∑k=0

a(µ,∗)k (x; γ)

(1− γ2)k+1 xkand B(n,∗)

µ (x; γ) =n∑

k=0

b(µ,∗)k (x; γ)

(1− γ2)k+1 xk.

For the case γ = 0.1 some of these differences are shown:

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x

........

-0.025........

0.025

.............4

.............6

.............8

.............10

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∆∗0(x, 0.1)

∆∗1(x, 0.1)

Figure 12 : Differences ∆∗0 (x; γ) and ∆∗

1 (x; γ) with γ = 0.1

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-0.015 ........

0.002

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......8......................10

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∆∗2(x, 0.1)

∆∗1(x, 0.1)

63


2 (x; γ) with γ = 0.1

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x

........0.0050

........

-0.0025 ........

0.0025

........12........6 ........8 ........10..............................................................................................................................................................................................................................

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∆∗2(x, 0.1)

∆∗3(x, 0.1)


3 (x; γ) with γ = 0.1

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........0.002

........0.001

........-0.001

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.........14

.........8

.........10

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∆∗3(x, 0.1)

∆∗4(x, 0.1)


4 (x; γ) with γ = 0.1

64

b) Integrals:

Holds (with 0 < β < α and β/α = γ < 1)∫J0(αx) J0(βx) dx =

1α

Θ0

(ax;

b

a

),

∫J1(αx) J1(βx) dx =

1α

Θ1

(ax;

b

a

)∫

I0(αx) I0(βx) dx =1α

Ω0

(ax;

b

a

),

∫I1(αx) I1(βx) dx =

1α

Ω1

(ax;

b

a

)(Θν and Ων as definded on pages 55 and 57.)

∫x2 · J0(αx) J0(βx) dx =

x(β2 + α2

)(α2 − β2)2

J0(αx) J0(βx)− β x2

α2 − β2J0(αx) J1(βx) +

α x2

α2 − β2J1(αx) J0(βx)+

+2xβ α

(α2 − β2)2J1(αx) J1(βx) +

β2 + α2

(α2 − β2)2

∫J0(αx) J0(βx) dx− 2

β α

(α2 − β2)2

∫J1(αx) J1(βx) dx

∫x2 · J1(αx) J1(βx) dx = 2

xβ α

(α2 − β2)2J0(αx) J0(βx)− α x2

α2 − β2J0(αx) J1(βx) +

β x2

α2 − β2J1(αx) J0(βx)+

+x(β2 + α2

)(α2 − β2)2

J1(αx) J1(βx) +2β α

(α2 − β2)2

∫J0(αx) J0(βx) dx− β2 + α2

(α2 − β2)2


∫x2 · I0(αx) I0(βx) dx = − (α2 + β2)x

(α2 − β2)2I0(αx) I0(βx)− β x2

α2 − β2I0(αx) I1(βx) +

α x2

α2 − β2I1(αx) I0(βx)+

+2 β αx

(α2 − β2)2I1(αx) I1(βx)− α2 + β2

(α2 − β2)2

∫I0(αx) I0(βx) dx− 2 β α

(α2 − β2)2

∫I1(αx) I1(βx) dx

∫x2 · I1(αx) I1(βx) dx =

2 xα β

(α2 − β2)2I0(αx) I0(βx) +

α x2

α2 − β2I0(αx) I1(βx)− β x2

α2 − β2I1(αx) I0(βx)−

− (α2 + β2)x(α2 − β2)2

I1(αx) I1(βx) +2 α β

(α2 − β2)2

∫I0(αx) I0(βx) dx +

α2 + β2

(α2 − β2)2


∫x4 · J0(αx) J0(βx) dx = 3

(x2α6 − 3 α4 − α4x2β2 − α2x2β4 − 10 α2β2 − 3 β4 + x2β6

)x

(α2 − β2)4J0(αx) J0(βx)−

−(x2α4 − 2 α2x2β2 − 15 α2 − 9 β2 + x2β4

)β x2

(α2 − β2)3J0(αx) J1(βx)+

+

(x2α4 − 2 α2x2β2 − 9 α2 − 15 β2 + x2β4

)α x2

(α2 − β2)3J1(αx) J0(βx)+

+6

(x2α4 − 2 α2x2β2 − 4 α2 − 4 β2 + x2β4

)β α x

(α2 − β2)4J1(αx) J1(βx)−

−9 α4 + 30 α2β2 + 9 β4

(α2 − β2)4

∫J0(αx) J0(βx) dx + 24

β α(β2 + α2

)(α2 − β2)4


∫x4 · J1(αx) J1(βx) dx = 6

(x2α4 − 4 α2 − 2 x2α2β2 + x2β4 − 4 β2

)β α x

(α2 − β2)4J0(αx) J0(βx)−

65

−(x2α4 − 2 α2x2β2 − 3 α2 − 21 β2 + x2β4

)α x2

(α2 − β2)3J0(αx) J1(βx)+

+

(x2α4 − 21 α2 − 2 α2x2β2 − 3 β2 + x2β4

)x2β

(α2 − β2)3J1(αx) J0(βx)+

+3

(x2α6 − α4 − x2α4β2 − 14 α2β2 − x2α2β4 − β4 + x2β6

)x

(α2 − β2)4J1(αx) J1(βx)−

−24α β

(α2 + β2

)(α2 − β2)4

∫J0(αx) J0(βx) dx +

3 α4 + 42 α2β2 + 3 β4

(α2 − β2)4


∫x4 · I0(αx) I0(βx) dx = −

3x[(α2 + β2

) (α2 − β2

)2x2 +

(β2 + 3 α2

) (3 β2 + α2

)]

(α2 − β2)4I0(αx) I0(βx)−

−βx2[

(α2 − β2

)2x2 + 15 α2 + 9 β2]

(α2 − β2)3I0(αx) I1(βx) +

α x2[(α2 − β2

)2x2 + 9 α2 + 15 β2]

(α2 − β2)3I1(αx) I0(βx)+

+6 α β x[

(α2 − β2

)2x2 + 4 α2 + 4 β2]

(α2 − β2)4I1(αx) I1(βx)−

−3(β2 + 3 α2

) (3 β2 + α2

)(α2 − β2)4

∫I0(αx) I0(βx) dx−

24 β α(α2 + β2

)(α2 − β2)4


∫x4 · I1(αx) I1(βx) dx =

6(α2 − β2

)2α β x3 + 24

(α2 + β2

)β αx

(α2 − β2)4I0(αx) I0(βx)+

+

(α2 − β2

)2α x4 + 3

(α2 + 7 β2

)α x2

(α2 − β2)3I0(αx) I1(βx)−

(α2 − β2

)2β x4 + 3

(7 α2 + β2

)β x2

(α2 − β2)3I1(αx) I0(βx)−

−3(α2 + β2

) (α2 − β2

)2x2 + 3 α4 + 42 β2α2 + 3 β4

(α2 − β2)4I1(αx) I1(βx)+

+24(α2 + β2

)β α

(α2 − β2)4


3 α4 + 42 β2α2 + 3 β4

(α2 − β2)4


Let ∫xn · Jν(αx) Jν(βx) dx =

P(ν)n (x)

(α2 − β2)nJ0(αx) J0(βx) +

Q(ν)n (x)

(α2 − β2)n−1J0(αx) J1(βx)+

+R

(ν)n (x)

(α2 − β2)n−1J1(αx) J0(βx) +

S(ν)n (x)

(α2 − β2)nJ1(αx) J1(βx)+

+U

(ν)n (x)

(α2 − β2)n


V(ν)n (x)

(α2 − β2)n


and ∫xn · Iν(αx) Iν(βx) dx =

P(ν)n (x)

(α2 − β2)nI0(αx) I0(βx) +

Q(ν)n (x)

(α2 − β2)n−1I0(αx) I1(βx)+

+R

(ν)n (x)

(α2 − β2)n−1I1(αx) I0(βx) +

S(ν)n (x)

(α2 − β2)nI1(αx) I1(βx)+

+U

(ν)n

(α2 − β2)n


V(ν)n

(α2 − β2)n

∫I1(αx) I1(βx) dx ,

then holds

P(0)6 = 5 x[

(β2 + α2

) (α2 − β2

)4x4 − 3

(5 α4 + 22 β2α2 + 5 β4

) (α2 − β2

)2x2+

66

+3(β2 + α2

) (15 β4 + 98 β2α2 + 15 α4

)]

Q(0)6 = −β x2[

(α2 − β2

)4x4 − 5

(11 α2 + 5 β2

) (α2 − β2

)2x2 + 465 α4 + 1230 β2α2 + 225 β4]

R(0)6 = α x2[

(α2 − β2

)4x4 − 5

(5 α2 + 11 β2

) (α2 − β2

)2x2 + 225 α4 + 1230 β2α2 + 465 β4]

S(0)6 = 10α β x[

(α2 − β2

)4x4 − 24

(β2 + α2

) (α2 − β2

)2x2 + 246 β2α2 + 69 α4 + 69 β4]

U(0)6 = 15

(β2 + α2

) (15 β4 + 98 β2α2 + 15 α4

)V

(0)6 = −30 β α

(23 β4 + 23 α4 + 82 β2α2

)P

(1)6 = 10α β x[

(α2 − β2

)4x4 − 24

(α2 + β2

) (α2 − β2

)2x2 + 81 α4 + 222 β2α2 + 81 β4]

Q(1)6 = −α x2[

(α2 − β2

)4x4 − 5

(3 α2 + 13 β2

) (α2 − β2

)2x2 + 45 α4 + 765 β4 + 1110 β2α2]

R(1)6 = x2β[

(α2 − β2

)4x4 − 5

(13 α2 + 3 β2

) (α2 − β2

)2x2 + 45 β4 + 765 α4 + 1110 β2α2]

S(1)6 = 5x[

(α2 + β2

) (α2 − β2

)4x4−

−3(3 α4 + 26 β2α2 + 3 β4

) (α2 − β2

)2x2 + 3

(α2 + β2

) (3 β4 + 122 β2α2 + 3 α4

)]

U(1)6 = 30

(27 α4 + 74 β2α2 + 27 β4

)β α

V(1)6 = −15

(α2 + β2

) (3 β4 + 122 β2α2 + 3 α4

)P

(0)6 = −5 x[

(β2 + α2

) (α2 − β2

)4x4 + 3

(5 α4 + 22 β2α2 + 5 β4

) (α2 − β2

)2x2+

+3(β2 + α2

) (15 β4 + 98 β2α2 + 15 α4

)]

Q(0)6 = −β x2[

(α2 − β2

)4x4 + 5

(11 α2 + 5 β2

) (α2 − β2

)2x2 + 465 α4 + 1230 β2α2 + 225 β4]

R(0)6 = α x2[

(α2 − β2

)4x4 + 5

(5 α2 + 11 β2

) (α2 − β2

)2x2 + 225 α4 + 1230 β2α2 + 465 β4]

S(0)6 = 10β α x[

(α2 − β2

)4x4 + 24

(β2 + α2

) (α2 − β2

)2x2 + 69 α4 + 69 β4 + 246 β2α2]

U(0)6 = −15

(β2 + α2

) (15 β4 + 98 β2α2 + 15 α4

)V

(0)6 = −30 β α

(23 β4 + 23 α4 + 82 β2α2

)P

(1)6 = 10α β x[

(α2 − β2

)4x4 + 24

(α2 + β2

) (α2 − β2

)2x2 + 222 β2α2 + 81 α4 + 81 β4]

Q(1)6 = α x2[

(α2 − β2

)4x4 + 5

(3 α2 + 13 β2

) (α2 − β2

)2x2 + 765 β4 + 45 α4 + 1110 β2α2]

R(1)6 = −x2β[

(α2 − β2

)4x4 + 5

(13 α2 + 3 β2

) (α2 − β2

)2x2 + 45 β4 + 765 α4 + 1110 β2α2]

S(1)6 = −5 x[

(α2 + β2

) (α2 − β2

)4x4 + 3

(3 α4 + 26 β2α2 + 3 β4

) (α2 − β2

)2x2+

+3(α2 + β2

) (3 β4 + 122 β2α2 + 3 α4

)]

U(1)6 = 30

(27 α4 + 74 β2α2 + 27 β4

)β α

V(1)6 = 15

(α2 + β2

) (3 β4 + 122 β2α2 + 3 α4

)

P(0)8 = 7x[

(α2 + β2

) (α2 − β2

)6x6 − 5

(7 α4 + 34 β2α2 + 7 β4

) (α2 − β2

)4x4+

+15(α2 + β2

) (35 β4 + 314 β2α2 + 35 α4

) (α2 − β2

)2x2−

−1575 β8 − 1575 α8 − 21540 α2β6 − 45930 α4β4 − 21540 α6β2]

Q(0)8 = −β x2[

(α2 − β2

)6x6 − 7

(17 α2 + 7 β2

) (α2 − β2

)4x4+

67

+35(107 α4 + 242 β2α2 + 35 β4

) (α2 − β2

)2x2−

−160755 α4β2 − 25935 α6 − 124845 α2β4 − 11025 β6]

R(0)8 = α x2[

(α2 − β2

)6x6 − 7

(7 α2 + 17 β2

) (α2 − β2

)4x4+

+35(35 α4 + 242 β2α2 + 107 β4

) (α2 − β2

)2x2−

−160755 α2β4 − 25935 β6 − 11025 α6 − 124845 α4β2]

S(0)8 = 14β α x[

(α2 − β2

)6x6 − 60

(α2 + β2

) (α2 − β2

)4x4+

+15(71 α4 + 242 β2α2 + 71 β4

) (α2 − β2

)2x2−

−240(α2 + β2

) (11 β4 + 74 β2α2 + 11 α4

)]

U(0)8 = −11025 α8 − 150780 α6β2 − 321510 α4β4 − 150780 α2β6 − 11025 β8

V(0)8 = 3360 α β

(α2 + β2

) (11 β4 + 74 β2α2 + 11 α4

)P

(1)8 = 14α β x[

(α2 − β2

)6x6 − 60

(α2 + β2

) (α2 − β2

)4x4+

+45(25 α4 + 78 β2α2 + 25 β4

) (α2 − β2

)2x2 − 720

(α2 + β2

) (5 β4 + 22 β2α2 + 5 α4

)]

Q(1)8 = −α x2[

(α2 − β2

)6x6 − 7

(5 α2 + 19 β2

) (α2 − β2

)4x4+

+105(3 β2 + 5 α2

) (15 β2 + α2

) (α2 − β2

)2x2 − 186165 β4α2 − 1575 α6 − 48825 β6 − 85995 β2α4]

R(1)8 = β x2[

(α2 − β2

)6x6 − 7

(5 β2 + 19 α2

) (α2 − β2

)4x4+

+105(5 β2 + 3 α2

) (β2 + 15 α2

) (α2 − β2

)2x2 − 85995 β4α2 − 48825 α6 − 1575 β6 − 186165 β2α4]

S(1)8 = 7 x[

(α2 + β2

) (α2 − β2

)6x6 − 5

(5 α4 + 38 β2α2 + 5 β4

) (α2 − β2

)4x4+

+45(α2 + β2

) (5 β4 + 118 β2α2 + 5 α4

) (α2 − β2

)2x2−

−19260 β2α6 − 225 β8 − 19260 β6α2 − 53190 β4α4 − 225 α8]

U(1)8 = −10080 β α

(α2 + β2

) (5 β4 + 22 β2α2 + 5 α4

)V

(1)8 = 1575 α8 + 134820 β2α6 + 372330 β4α4 + 134820 β6α2 + 1575 β8

P(0)8 = −7 x[

(α2 + β2

) (α2 − β2

)6x6 + 5

(7 α4 + 34 β2α2 + 7 β4

) (α2 − β2

)4x4+

+15(α2 + β2

) (35 β4 + 314 β2α2 + 35 α4

) (α2 − β2

)2x2+

+21540β2α6 + 1575 β8 + 21540β6α2 + 45930 β4α4 + 1575 α8]

Q(0)8 = −β x2[

(α2 − β2

)6x6 + 7

(17 α2 + 7 β2

) (α2 − β2

)4x4+

+35(107 α4 + 242 β2α2 + 35 β4

) (α2 − β2

)2x2+

+160755β2α4 + 11025 β6 + 124845 β4α2 + 25935 α6]

R(0)8 = α x2[

(α2 − β2

)6x6 + 7

(7 α2 + 17 β2

) (α2 − β2

)4x4+

+35(35 α4 + 242 β2α2 + 107 β4

) (α2 − β2

)2x2+

+124845β2α4 + 11025α6 + 160755 β4α2 + 25935 β6]

S(0)8 = 14α β x[

(α2 − β2

)6x6 + 60

(α2 + β2

) (α2 − β2

)4x4+

+15(71 α4 + 242 β2α2 + 71 β4

) (α2 − β2

)2x2+

+240(α2 + β2

) (11 β4 + 74 β2α2 + 11 α4

)]

U(0)8 = −11025 α8 − 150780 β2α6 − 321510 β4α4 − 150780 β6α2 − 11025 β8

68

V(0)8 = −3360 β α

(α2 + β2

) (11 β4 + 74 β2α2 + 11 α4

)P

(1)8 = 14β α x[

(α2 − β2

)6x6 + 60

(α2 + β2

) (α2 − β2

)4x4+

+45(25 α4 + 78 β2α2 + 25 β4

) (α2 − β2

)2x2 + 720

(α2 + β2

) (5 β4 + 22 β2α2 + 5 α4

)]

Q(1)8 = α x2[

(α2 − β2

)6x6 + 7

(5 α2 + 19 β2

) (α2 − β2

)4x4+

+105(3 β2 + 5 α2

) (15 β2 + α2

) (α2 − β2

)2x2 + 85995 β2α4 + 1575 α6 + 186165 β4α2 + 48825 β6]

R(1)8 = −β x2[

(α2 − β2

)6x6 + 7

(19 α2 + 5 β2

) (α2 − β2

)4x4+

+105(5 β2 + 3 α2

) (β2 + 15 α2

) (α2 − β2

)2x2 + 85995 β4α2 + 1575 β6 + 186165 β2α4 + 48825 α6]

S(1)8 = −7 x[

(α2 + β2

) (α2 − β2

)6x6 + 5

(5 α4 + 38 β2α2 + 5 β4

) (α2 − β2

)4x4+

+45(α2 + β2

) (5 β4 + 118 β2α2 + 5 α4

) (α2 − β2

)2x2+

+19260β2α6 + 225 β8 + 19260 β6α2 + 53190β4α4 + 225 α8]

U(1)8 = 10080

(α2 + β2

) (5 β4 + 22 β2α2 + 5 α4

)α β

V(1)8 = 1575 α8 + 134820 β2α6 + 372330 β4α4 + 134820 β6α2 + 1575 β8

P(0)10 = 9 x[

(α2 + β2

) (α2 − β2

)8x8 − 7

(9 α4 + 46 β2α2 + 9 β4

) (α2 − β2

)6x6+

+105(α2 + β2

) (21 β4 + 214 β2α2 + 21 α4

) (α2 − β2

)4x4−

−315(β2 + 15 α2

) (15 β2 + α2

) (7 β4 + 18 β2α2 + 7 α4

) (α2 − β2

)2x2+

+315(α2 + β2

) (315 β8 + 6548 α2β6 + 19042 α4β4 + 6548 α6β2 + 315 α8

)]

Q(0)10 = −β x2[

(α2 − β2

)8x8 − 9

(9 β2 + 23 α2

) (α2 − β2

)6x6+

+63(223 α4 + 482 α2β2 + 63 β4

) (α2 − β2

)4x4−

−945(391 α6 + 2139 β2α4 + 1461 β4α2 + 105 β6

) (α2 − β2

)2x2+

+25957260β2α6 + 2299185 α8 + 46590390 α4β4 + 17157420 α2β6 + 893025 β8]

R(0)10 = α x2[

(α2 − β2

)8x8 − 9

(9 α2 + 23 β2

) (α2 − β2

)6x6+

+63(63 α4 + 482 α2β2 + 223 β4

) (α2 − β2

)4x4−

−945(105 α6 + 1461 β2α4 + 2139 β4α2 + 391 β6

) (α2 − β2

)2x2+

+17157420β2α6 + 893025 α8 + 46590390 α4β4 + 25957260α2β6 + 2299185β8]

S(0)10 = 18 β α x[

(α2 − β2

)8x8 − 112

(α2 + β2

) (α2 − β2

)6x6+

+35(143 α4 + 482 α2β2 + 143 β4

) (α2 − β2

)4x4−

−2520(α2 + β2

) (31 β4 + 194 α2β2 + 31 α4

) (α2 − β2

)2x2+

+2395260β2α6 + 177345 α8 + 5176710α4β4 + 2395260 α2β6 + 177345 β8]

U(0)10 = 2835

(α2 + β2

) (315 β8 + 6548 α2β6 + 19042α4β4 + 6548 β2α6 + 315 α8

)V

(0)10 = −5670 β α

(563 β8 + 563 α8 + 7604 β2α6 + 16434 α4β4 + 7604 α2β6

)P

(1)10 = 18xα β[

(α2 − β2

)8x8 − 112

(α2 + β2

) (α2 − β2

)6x6+

+105(49 α4 + 158 β2α2 + 49 β4

) (α2 − β2

)4x4−

69

−2520(α2 + β2

) (35 β4 + 186 β2α2 + 35 α4

) (α2 − β2

)2x2+

+2485980β2α6 + 275625 β8 + 2485980 β6α2 + 4798710β4α4 + 275625 α8]

Q(1)10 = −α x2[

(α2 − β2

)8x8 − 9

(7 α2 + 25 β2

) (α2 − β2

)6x6+

+63(35 α4 + 474 β2α2 + 259 β4

) (α2 − β2

)4x4−

−945(35 α6 + 1223 β2α4 + 2313 β4α2 + 525 β6

) (α2 − β2

)2(β + α)2 x2+

+9718380β2α6 + 4862025 β8 + 35029260 β6α2 + 43188390 β4α4 + 99225 α8]

R(1)10 = β x2[

(α2 − β2

)8x8 − 9

(7 β2 + 25 α2

) (α2 − β2

)6x6+

+63(259 α4 + 474 β2α2 + 35 β4

) (α2 − β2

)4x4−

−945(525 α6 + 2313 β2α4 + 1223 β4α2 + 35 β6

) (α2 − β2

)2x2+

+35029260β2α6 + 99225 β8 + 9718380β6α2 + 43188390β4α4 + 4862025α8]

S(1)10 = 9x[

(α2 + β2

) (α2 − β2

)8x8 − 7

(7 β2 + α2

) (β2 + 7 α2

) (α2 − β2

)6x6+

+35(α2 + β2

) (35 β4 + 698 β2α2 + 35 α4

) (α2 − β2

)4x4−

−315(35 α8 + 1748 β2α6 + 4626 β4α4 + 1748 β6α2 + 35 β8

) (α2 − β2

)2x2+

+315(α2 + β2

) (35 β8 + 5108 β6α2 + 22482 β4α4 + 5108 β2α6 + 35 α8

)]

U(1)10 = 5670

(875 α8 + 7892 β2α6 + 15234 β4α4 + 7892 β6α2 + 875 β8

)β α

V(1)10 = −2835

(α2 + β2

) (35 β8 + 5108 β6α2 + 22482 β4α4 + 5108 β2α6 + 35 α8

)P

(0)10 = −9 x[

(β2 + α2

) (α2 − β2

)8x8 + 7

(9 α4 + 46 β2α2 + 9 β4

) (α2 − β2

)6x6+

+105(β2 + α2

) (21 β4 + 214 β2α2 + 21 α4

) (α2 − β2

)4x4+

+315(β2 + 15 α2

) (15 β2 + α2

) (7 β4 + 18 β2α2 + 7 α4

) (α2 − β2

)2x2+

+315(β2 + α2

) (315 β8 + 6548 β6α2 + 19042 β4α4 + 6548 β2α6 + 315 α8

)]

Q(0)10 = −β x2[

(α2 − β2

)8x8 + 9

(23 α2 + 9 β2

) (α2 − β2

)6x6+

+63(223 α4 + 482 β2α2 + 63 β4

) (α2 − β2

)4x4+

+945(391 α6 + 2139 α4β2 + 1461 β4α2 + 105 β6

) (α2 − β2

)2x2+

+46590390β4α4 + 25957260 β2α6 + 17157420 β6α2 + 2299185α8 + 893025 β8]

R(0)10 = α x2[

(α2 − β2

)8x8 + 9

(9 α2 + 23 β2

) (α2 − β2

)6x6+

+63(63 α4 + 482 β2α2 + 223 β4

) (α2 − β2

)4x4+

+945(105 α6 + 1461 β2α4 + 2139 β4α2 + 391 β6

) (α2 − β2

)2x2+

+2299185β8 + 893025 α8 + 46590390β4α4 + 25957260 β6α2 + 17157420 β2α6]

S(0)10 = 18 β α x[

(α2 − β2

)8x8 + 112

(α2 + β2

) (α2 − β2

)6x6+

+35(143 α4 + 482 β2α2 + 143 β4

) (α2 − β2

)4x4+

+2520(α2 + β2

) (31 β4 + 194 β2α2 + 31 α4

) (α2 − β2

)2x2+

+177345β8 + 177345 α8 + 5176710β4α4 + 2395260β6α2 + 2395260β2α6]

U(0)10 = −2835

(α2 + β2

) (315 β8 + 6548 β6α2 + 19042 β4α4 + 6548 β2α6 + 315 α8

)V

(0)10 = −5670 β α

(16434 β4α4 + 7604 β2α6 + 563 α8 + 7604 β6α2 + 563 β8

)70

P(1)10 = 18 α β x[

(α2 − β2

)8x8 + 112

(α2 + β2

) (α2 − β2

)6x6+

+105(49 α4 + 158 β2α2 + 49 β4

) (α2 − β2

)4x4+

+2520(α2 + β2

) (35 β4 + 186 β2α2 + 35 α4

) (α2 − β2

)2(β + α)2 x2+

+2485980β2α6 + 275625 β8 + 2485980β6α2 + 4798710β4α4 + 275625 α8]

Q(1)10 = α x2[

(α2 − β2

)8x8 + 9

(7 α2 + 25 β2

) (α2 − β2

)6(β + α)6 x6+

+63(35 α4 + 474 β2α2 + 259 β4

) (α2 − β2

)4x4+

+945(35 α6 + 1223 β2α4 + 2313 β4α2 + 525 β6

) (α2 − β2

)2x2+

+9718380β2α6 + 4862025β8 + 35029260 β6α2 + 43188390 β4α4 + 99225 α8]

R(1)10 = −β x2[

(α2 − β2

)8x8 + 9

(7 β2 + 25 α2

) (α2 − β2

)6(β + α)6 x6+

+63(259 α4 + 474 β2α2 + 35 β4

) (α2 − β2

)4x4+

+945(525 α6 + 2313 β2α4 + 1223 β4α2 + 35 β6

) (α2 − β2

)2x2+

+35029260β2α6 + 99225β8 + 9718380 β6α2 + 43188390 β4α4 + 4862025α8]

S(1)10 = −9 x[

(α2 + β2

) (α2 − β2

)8x8 + 7

(7 β2 + α2

) (β2 + 7 α2

) (α2 − β2

)6x6+

+35(α2 + β2

) (35 β4 + 698 β2α2 + 35 α4

) (α2 − β2

)4x4+

+315(35 α8 + 1748 β2α6 + 4626 β4α4 + 1748 β6α2 + 35 β8

) (α2 − β2

)2x2+

+315(α2 + β2

) (35 β8 + 5108 β6α2 + 22482 β4α4 + 5108 β2α6 + 35 α8

)]

U(1)10 = 5670

(875 α8 + 7892 β2α6 + 15234β4α4 + 7892 β6α2 + 875 β8

)β α

V(1)10 = 2835

(α2 + β2

) (35 β8 + 5108 β6α2 + 22482 β4α4 + 5108 β2α6 + 35 α8

)

71

XVI. Integrals of the type∫

x2n+1 · Z0(αx) · Z1(βx) dx

Holds (with 0 < β < α and β/α = γ < 1)∫J0(αx) J0(βx) dx =

1α

Θ0

(ax;

b

a

),

∫J1(αx) J1(βx) dx =

1α

Θ1

(ax;

b

a

),


1α

Ω0

(ax;

b

a

),


1α

Ω1

(ax;

b

a

).

(Θν and Ων as definded on pages 55 and 57. In these integrals both Bessel functions are of the same order,so one can suppose β < α. This relation is not presumed for the product Z0(αx) · Z1(βx), that means forthe following integrals.)∫

J0(αx) J1(βx) dx

x= −J0(αx) J1(βx) + β

∫J0(αx) J0(βx) dx− α

∫J1(αx) J1(βx) dx∫

I0(αx) I1(βx) dx

x= −I0(αx) I1(βx) + β

∫I0(αx) I1(βx) dx + α


∫xJ0(αx) J1(βx) dx =

=x

α2 − β2[β J0(αx) J0(βx) + α J1(αx) J1(βx)]− β

α2 − β2

∫J0(αx) J0(βx) dx+

α

α2 − β2

∫J1(αx) J1(βx) dx∫

x I0(αx) I1(βx) dx =

=x

α2 − β2[α I1(αx) I1(βx)− β I0(αx) I0(βx)]+

β

α2 − β2

∫I0(αx) I0(βx) dx+

α

α2 − β2


Let ∫xn · J0(αx) J1(βx) dx =

Pn(x)(α2 − β2)n

J0(αx) J0(βx) +Qn(x)

(α2 − β2)n−1J0(αx) J1(βx)+

+Rn(x)

(α2 − β2)n−1J1(αx) J0(βx) +

Sn(x)(α2 − β2)n

J1(αx) J1(βx)+

+Un

(α2 − β2)n


Vn

(α2 − β2)n


and ∫xn · I0(αx) I1(βx) dx =

Pn(x)(α2 − β2)n

I0(αx) I0(βx) +Qn(x)

(α2 − β2)n−1I0(αx) I1(βx)+

+Rn(x)

(α2 − β2)n−1I1(αx) I0(βx) +

Sn(x)(α2 − β2)n

I1(αx) I1(βx)+

+Un

(α2 − β2)n


Vn

(α2 − β2)n

∫I1(αx) I1(βx) dx ,

then holds

P(0)3 = β x[

(α2 − β2

)2x2 − 3 β2 − 5 α2] , Q

(0)3 = x2[α2 + 3 β2]

R(0)3 = −4 α β x2 , S

(0)3 = α x[

(α2 − β2

)2x2 − α2 − 7 β2]

U(0)3 = −β

(5 α2 + 3 β2

), V

(0)3 = α

(7 β2 + α2

)P

(0)3 = −β x[

(α2 − β2

)2x2 + 5 α2 + 3 β2] , Q

(0)3 = −[α2 + 3 β2]x2

R(0)3 = 4α β x2 , S

(0)3 = α x

(α2 − β2

)2x2 + 7 β2 + α2

U(0)3 = −β

(5 α2 + 3 β2

), V

(0)3 = −α

(7 β2 + α2

)72

P(0)5 = β x[

(α2 − β2

)4x4 − 3

(11 α2 + 5 β2

) (α2 − β2

)2x2 + 3

(3 β2 + 13 α2

) (3 α2 + 5 β2

)]

Q(0)5 = x2[

(3 α2 + 5 β2

) (α2 − β2

)2x2 − 3

(15 β2 + α2

) (β2 + 3 α2

)]

R(0)5 = −4 β α x2[2

(α2 − β2

)2x2 − 27 α2 − 21 β2]

S(0)5 = α x[

(α2 − β2

)4x4 − 3

(3 α2 + 13 β2

) (α2 − β2

)2x2 + 9 α4 + 246 α2β2 + 129 β4]

U(0)5 = 3β

(3 β2 + 13 α2

) (3 α2 + 5 β2

), V

(0)5 = −3 α

(43 β4 + 82 α2β2 + 3 α4

)

P(0)5 = −β x[

(α2 − β2

)4x4 + 3

(11 α2 + 5 β2

) (α2 − β2

)2x2 + 3

(3 β2 + 13 α2

) (3 α2 + 5 β2

)]

Q(0)5 = −x2[

(3 α2 + 5 β2

) (α2 − β2

)2x2 + 3

(15 β2 + α2

) (β2 + 3 α2

)]

R(0)5 = 4 β α x2[2

(α2 − β2

)2x2 + 27 α2 + 21 β2]

S(0)5 = α x[

(α2 − β2

)4x4 + 3

(3 α2 + 13 β2

) (α2 − β2

)2x2 + 9 α4 + 129 β4 + 246 β2α2]

U(0)5 = −3

(3 β2 + 13 α2

) (3 α2 + 5 β2

)β

V(0)5 = −3 α

(43 β4 + 82 β2α2 + 3 α4

)P

(0)7 = β x[

(α2 − β2

)6x6 − 5

(17 α2 + 7 β2

) (α2 − β2

)4x4+

+15(115 α4 + 234 β2α2 + 35 β4

) (α2 − β2

)2x2 − 1575 β6 − 5625 α6 − 15915 β4α2 − 22965 β2α4]

Q(0)7 = x2[

(5 α2 + 7 β2

) (α2 − β2

)4x4 − 5

(15 α4 + 142 β2α2 + 35 β4

) (α2 − β2

)2x2+

+1575β6 + 8805 β2α4 + 225 α6 + 12435β4α2]

R(0)7 = −4 β α x2[3

(α2 − β2

)4x4 − 5

(25 α2 + 23 β2

) (α2 − β2

)2x2 + 1350 α4 + 3540 β2α2 + 870 β4]

S(0)7 = α x[

(α2 − β2

)6x6 − 5

(19 β2 + 5 α2

) (α2 − β2

)4x4+

+15(15 α4 + 242 β2α2 + 127 β4

) (α2 − β2

)2x2 − 14205 β2α4 − 5055 β6 − 26595 β4α2 − 225 α6]

U(0)7 = −15

(375 α6 + 1531 β2α4 + 1061 β4α2 + 105 β6

)β

V(0)7 = 15 α

(947 β2α4 + 15 α6 + 337 β6 + 1773 β4α2

)P

(0)7 = −β x[

(α2 − β2

)6x6 + 5

(17 α2 + 7 β2

) (α2 − β2

)4x4+

+15(115 α4 + 234 β2α2 + 35 β4

) (α2 − β2

)2x2 + 1575 β6 + 5625 α6 + 15915 β4α2 + 22965 β2α4]

Q(0)7 = −x2[

(5 α2 + 7 β2

) (α2 − β2

)4x4 + 5

(15 α4 + 142 β2α2 + 35 β4

) (α2 − β2

)2x2+

+225α6 + 8805 β2α4 + 12435 β4α2 + 1575 β6]

R(0)7 = 4β α x2[3

(α2 − β2

)4x4 + 5

(25 α2 + 23 β2

) (α2 − β2

)2x2 + 1350 α4 + 3540 β2α2 + 870 β4]

S(0)7 = α x[

(α2 − β2

)6x6 + 5

(19 β2 + 5 α2

) (α2 − β2

)4x4+

+15(15 α4 + 242 β2α2 + 127 β4

) (α2 − β2

)2x2 + 14205 β2α4 + 225 α6 + 26595 β4α2 + 5055 β6]

U(0)7 = −15

(375 α6 + 1531 β2α4 + 1061 β4α2 + 105 β6

)β

V(0)7 = −15 α

(947 β2α4 + 15 α6 + 337 β6 + 1773 β4α2

)

P(0)9 = β x[

(α2 − β2

)8x8−7

(23 α2 + 9 β2

) (α2 − β2

)6x6+105

(β2 + 7 α2

) (21 β2 + 11 α2

) (α2 − β2

)4x4−

73

−315(455 α6 + 2139 β2α4 + 1397 β4α2 + 105 β6

) (α2 − β2

)2x2+

+3262140β2α6 + 452025 α8 + 4798710β4α4 + 1709820β6α2 + 99225β8]

Q(0)9 = x2[

(9 β2 + 7 α2

) (α2 − β2

)6x6 − 7

(35 α4 + 286 β2α2 + 63 β4

) (α2 − β2

)4x4+

+105(35 α6 + 867 β2α4 + 1041 β4α2 + 105 β6

) (α2 − β2

)2x2−

−2749950 β4α4 − 99225 β8 − 835380 β2α6 − 11025 α8 − 1465380 β6α2]

R(0)9 = −4 β α x2[4

(α2 − β2

)6x6 − 7

(49 α2 + 47 β2

) (α2 − β2

)4x4+

+105(105 α4 + 318 β2α2 + 89 β4

) (α2 − β2

)2x2 − 630

(β2 + 7 α2

) (97 β4 + 134 β2α2 + 25 α4

)]

S(0)9 = α x[

(α2 − β2

)8x8−7

(7 α2 + 25 β2

) (α2 − β2

)6x6 +35

(35 α4 + 482 β2α2 + 251 β4

) (α2 − β2

)4x4−

−315(35 α6 + 1287 β2α4 + 2313 β4α2 + 461 β6

) (α2 − β2

)2x2+

+343665β8 + 5176710β4α4 + 3514140β6α2 + 1276380 β2α6 + 11025 α8]

U(0)9 = 315β

(15234 β4α4 + 5428 β6α2 + 315 β8 + 1435 α8 + 10356 β2α6

)V

(0)9 = −315

(35 α8 + 4052 β2α6 + 16434β4α4 + 11156 β6α2 + 1091 β8

)α

P(0)9 = −β x[

(α2 − β2

)8x8+7

(23 α2 + 9 β2

) (α2 − β2)

)6x6+105

(β2 + 7 α2

) (21 β2 + 11 α2

) (α2 − β2

)4x4+

+315(455 α6 + 2139 β2α4 + 1397 β4α2 + 105 β6

) (α2 − β2

)2x2+

+1709820β6α2 + 3262140 β2α6 + 99225 β8 + 452025 α8 + 4798710 β4α4]

Q(0)9 = −x2[

(7 α2 + 9 β2

) (α2 − β2

)6x6 + 7

(35 α4 + 286 β2α2 + 63 β4

) (α2 − β2

)4x4+

+105(35 α6 + 867 β2α4 + 1041 β4α2 + 105 β6

) (α2 − β2

)2x2+

+99225β8 + 1465380 β6α2 + 2749950 β4α4 + 835380 β2α6 + 11025α8]

R(0)9 = 4β α x2[4

(α2 − β2

)6x6 + 7

(49 α2 + 47 β2

) (α2 − β2

)4x4+

+105(105 α4 + 318 β2α2 + 89 β4

) (α2 − β2

)2x2 + 630

(β2 + 7 α2

) (97 β4 + 134 β2α2 + 25 α4

)]

S(0)9 = α x[

(α2 − β2

)8x8 +7

(7 α2 + 25 β2

) (α2 − β2

)6x6 +35

(35 α4 + 482 β2α2 + 251 β4

) (α2 − β2

)4x4+

+315(35 α6 + 1287 β2α4 + 2313 β4α2 + 461 β6

) (α2 − β2

)2x2+

+3514140β6α2 + 5176710β4α4 + 1276380β2α6 + 343665 β8 + 11025 α8]

U(0)9 = −315

(1435 α8 + 10356 β2α6 + 15234β4α4 + 5428 β6α2 + 315 β8

)β

V(0)9 = −315 α

(35 α8 + 4052 β2α6 + 16434β4α4 + 11156β6α2 + 1091 β8

)P

(0)11 = β x[

(α2 − β2

)10x10 − 9

(29 α2 + 11 β2

) (α2 − β2

)8x8+

+63(387 α4 + 794 β2α2 + 99 β4

) (α2 − β2

)6x6−

−945(1113 α6 + 5429 β2α4 + 3467 β4α2 + 231 β6

) (α2 − β2

)4x4+

+2835(6195 α8 + 52196 β2α6 + 78882 β4α4 + 25412 β6α2 + 1155 β8

) (α2 − β2

)2x2−

−54474525 α10 − 1575415170 β4α6 − 1200752910 β6α4 − 9823275 β10 − 258673905β8α2 − 616751415 β2α8]

Q(0)11 = x2[

(9 α2 + 11 β2

) (α2 − β2

)8x8 − 9

(63 α4 + 478 β2α2 + 99 β4

) (α2 − β2

)6x6+

+63(315 α6 + 6719 β2α4 + 7633 β4α2 + 693 β6

) (α2 − β2

)4x4−

−945(315 α8 + 15308 β2α6 + 44346β4α4 + 20796 β6α2 + 1155 β8

) (α2 − β2

)2x2+

74

+893025α10 + 674225370β4α6 + 232489845β8α2 + 112756455β2α8 + 827757630β6α4 + 9823275β10]

R(0)11 = −4 β α x2[5

(α2 − β2

)8x8 − 9

(81 α2 + 79 β2

) (α2 − β2

)6x6+

+252(189 α4 + 598 β2α2 + 173 β4

) (α2 − β2

)4x4−

−1890(735 α6 + 4611 β2α4 + 4317 β4α2 + 577 β6

) (α2 − β2

)2x2+

+125998740β2α6 + 6546015 β8 + 93248820 β6α2 + 225297450β4α4 + 13395375α8]

S(0)11 = α x[

(α2 − β2

)10x10 − 9

(9 α2 + 31 β2

) (α2 − β2)

)8x8+

+63(63 α4 + 802 β2α2 + 415 β4

) (α2 − β2

)6x6−

−315(315 α6 + 9743 β2α4 + 17201 β4α2 + 3461 β6

) (α2 − β2

)4x4+

+2835(315 α8 + 21188 β2α6 + 81234 β4α4 + 55332 β6α2 + 5771 β8

) (α2 − β2

)2x2−

−166337955 β2α8 − 1728947430 β6α4 − 893025 α10 − 605485125 β8α2 − 1178220330 β4α6 − 36007335 β10]

U(0)11 = −2835

(19215 α10 + 217549 β2α8 + 555702 β4α6 + 423546 β6α4 + 91243 β8α2 + 3465 β10

)β

V(0)11 = 2835 α

(315 α10 + 58673 β2α8 + 415598 β4α6 + 609858 β6α4 + 213575 β8α2 + 12701 β10

)P

(0)11 = −β x[

(α2 − β2

)10x10 + 9

(29 α2 + 11 β2

) (α2 − β2

)8x8+

+63(387 α4 + 794 β2α2 + 99 β4

) (α2 − β2

)6x6+

+945(1113 α6 + 5429 β2α4 + 3467 β4α2 + 231 β6

) (α2 − β2

)4x4+

+2835(6195 α8 + 52196β2α6 + 78882β4α4 + 25412 β6α2 + 1155 β8

) (α2 − β2

)2x2+

+258673905β8α2 + 54474525 α10 + 1575415170 β4α6 + 616751415 β2α8 + 1200752910 β6α4 + 9823275 β10]

Q(0)11 = −x2[

(9 α2 + 11 β2

) (α2 − β2

)8x8 + 9

(63 α4 + 478 β2α2 + 99 β4

) (α2 − β2

)6x6+

+63(315 α6 + 6719 β2α4 + 7633 β4α2 + 693 β6

) (α2 − β2

)4x4+

+945(315 α8 + 15308 β2α6 + 44346 β4α4 + 20796β6α2 + 1155 β8

) (α2 − β2

)2x2+

+893025α10 + 674225370β4α6 + 232489845β8α2 + 9823275 β10 + 112756455β2α8 + 827757630β6α4]

R(0)11 = 4 β α x2[

(α2 − β2

)8x8 + 9

(81 α2 + 79 β2

) (α2 − β2

)6x6+

+252(189 α4 + 598 β2α2 + 173 β4

) (α2 − β2

)4x4+

+1890(735 α6 + 4611 β2α4 + 4317 β4α2 + 577 β6

) (α2 − β2

)2x2+

+93248820β6α2 + 225297450β4α4 + 125998740β2α6 + 6546015β8 + 13395375 α8]

S(0)11 = α x[

(α2 − β2

)10x10+9

(9 α2 + 31 β2

) (α2 − β2

)8x8+63

(63 α4 + 802 β2α2 + 415 β4

) (α2 − β2

)6x6+

+315(315 α6 + 9743 β2α4 + 17201 β4α2 + 3461 β6

) (α2 − β2

)4x4+

+2835(315 α8 + 21188β2α6 + 81234β4α4 + 55332 β6α2 + 5771 β8

) (α2 − β2

)2x2+

+893025α10 + 1178220330β4α6 + 166337955β2α8 + 1728947430 β6α4 + 36007335 β10 + 605485125β8α2]

U(0)11 = −2835

(19215 α10 + 217549 β2α8 + 555702 β4α6 + 423546 β6α4 + 91243β8α2 + 3465 β10

)β

V(0)11 = −2835 α

(315 α10 + 58673 β2α8 + 415598 β4α6 + 609858 β6α4 + 213575 β8α2 + 12701 β10

)

75

XVII. Integrals of the type∫

x2n+1 · J0(x) · I0(x) dx

∫x · J0(x) · I0(x) dx =

x

2[J0(x) · I1(x) + J1(x) · I0(x)]∫

x3 · J0(x) · I0(x) dx =12[x3J0(x) · I1(x) + x3J1(x) · I0(x)− 2x2J1(x) · I1(x)

]∫

x5 · J0(x) · I0(x) dx =

=12[8 x2J0(x) · I0(x) + (x5 − 4 x3 − 8 x)J0(x) · I1(x) + (x5 + 4 x3 − 8 x)J1(x) · I0(x)− 4 x4J1(x) · I1(x)

]∫

x7 · J0(x) · I0(x) dx =12[48 x4J0(x) · I0(x) + (x7 − 12 x5 − 96 x3)J0(x) · I1(x)+

+(x7 + 12 x5 − 96 x3)J1(x) · I0(x) + (−6 x6 + 192 x2)J1(x) · I1(x)]∫

x9 · J0(x) · I0(x) dx =

=12[(144 x6 − 3456 x2)J0(x) · I0(x) + (x9 − 24 x7 − 432 x5 + 1728 x3 + 3456 x)J0(x) · I1(x)+

+(x9 + 24 x7 − 432 x5 − 1728 x3 + 3456 x)J1(x) · I0(x) + (−8 x8 + 1728 x4)J1(x) · I1(x)]

With∫xn·J0(x)·I0(x) dx =

12

[Pn(x)J0(x) · I0(x) + Qn(x)J0(x) · I1(x) + Rn(x)J1(x) · I0(x) + Sn(x)J1(x) · I1(x)]

holdsP11(x) = 320 x8 − 61440 x4

Q11(x) = x11 − 40 x9 − 1280 x7 + 15360x5 + 122880 x3

R11(x) = x11 + 40 x9 − 1280 x7 − 15360 x5 + 122880 x3

S11(x) = −10 x10 + 7680 x6 − 245760 x2

P13(x) = 600 x10 − 432000 x6 + 10368000 x2

Q13(x) = x13 − 60 x11 − 3000 x9 + 72000 x7 + 1296000x5 − 5184000 x3 − 10368000 x

R13(x) = x13 + 60 x11 − 3000 x9 − 72000 x7 + 1296000x5 + 5184000x3 − 10368000 x

S13(x) = −12 x12 + 24000 x8 − 5184000x4

P15(x) = 1008 x12 − 1935360x8 + 371589120x4

Q15(x) = x15 − 84 x13 − 6048 x11 + 241920 x9 + 7741440 x7 − 92897280 x5 − 743178240 x3

R15(x) = x15 + 84 x13 − 6048 x11 − 241920 x9 + 7741440 x7 + 92897280 x5 − 743178240 x3

S15(x) = −14 x14 + 60480 x10 − 46448640 x6 + 1486356480x2

76

XVIII. Integrals of the type∫

x2n · J0(x) · I1(x) dx∫x2 · J0(x) · I1(x) dx =

12[x2J0(x) · I0(x)− xJ0(x) · I1(x)− xJ1(x) · I0(x) + x2J1(x) · I1(x)

]∫

x4 · J0(x) · I1(x) dx =

=12[(x4 − 2 x2)J0(x) · I0(x) + (−x3 + 2 x)J0(x) · I1(x) + (−3 x3 + 2 x)J1(x) · I0(x) + (x4 + 4 x2)J1(x) · I1(x)

]With∫

xn·J0(x)·I1(x) dx =12


holdsP6(x) = x6 − 8 x4 − 24 x2 , Q6(x) = −x5 + 28 x3 + 24 x

R6(x) = −5 x5 + 4 x3 + 24 x , S6(x) = x6 + 12 x4 − 32 x2

P8(x) = x8 − 18 x6 − 192 x4 + 432 x2 , Q8(x) = −x7 + 102 x5 + 168 x3 − 432 x

R8(x) = −7 x7 + 6 x5 + 600 x3 − 432 x , S8(x) = x8 + 24 x6 − 216 x4 − 768 x2

P10(x) = x10 − 32 x8 − 720 x6 + 6144 x4 + 17280x2

Q10(x) = −x9 + 248 x7 + 624 x5 − 20928 x3 − 17280 x

R10(x) = −9 x9 + 8 x7 + 3696 x5 − 3648 x3 − 17280 x

S10(x) = x10 + 40 x8 − 768 x6 − 8640 x4 + 24576 x2

P12(x) = x12 − 50 x10 − 1920 x8 + 36000 x6 + 368640 x4 − 864000 x2

Q12(x) = −x11 + 490 x9 + 1680 x7 − 200160 x5 − 305280 x3 + 864000 x

R12(x) = −11 x11 + 10 x9 + 13680 x7 − 15840 x5 − 1169280x3 + 864000 x

S12(x) = x12 + 60 x10 − 2000 x8 − 46080 x6 + 432000 x4 + 1474560x2

P14(x) = x14 − 72 x12 − 4200 x10 + 138240 x8 + 3024000x6 − 26542080 x4 − 72576000 x2

Q14(x) = −x13 + 852 x11 + 3720 x9 − 1056960x7 − 2436480 x5 + 89372160 x3 + 72576000x

R14(x) = −13 x13 + 12 x11 + 38280x9 − 48960 x7 − 15707520 x5 + 16796160 x3 + 72576000 x

S14(x) = x14 + 84 x12 − 4320 x10 − 168000 x8 + 3317760 x6 + 36288000 x4 − 106168320 x2

P16(x) =

= x16 − 98 x14 − 8064 x12 + 411600 x10 + 15482880x8 − 296352000x6 − 2972712960 x4 + 7112448000x2

Q16(x) =

= −x15 +1358x13 +7224x11− 3993360 x9− 12539520 x7 +1632234240x5 +2389201920x3− 7112448000 x

R16(x) =

= −15 x15 +14 x13 +89544x11−122640 x9−111323520 x7 +145877760x5 +9501649920x3−7112448000 x

S16(x) =

= x16 + 112 x14 − 8232 x12 − 483840 x10 + 16464000x8 + 371589120x6 − 3556224000 x4 − 11890851840x2

77

XIX. Integrals of the type∫

x2n · J1(x) · I0(x) dx∫x2 · J1(x) · I0(x) dx =

12[−x2J0(x) · I0(x) + xJ0(x) · I1(x) + xJ1(x) · I0(x) + x2J1(x) · I1(x)

]∫

x4 · J1(x) · I0(x) dx =

=12[(−x4 − 2 x2)J0(x) · I0(x) + (3x3 + 2 x)J0(x) · I1(x) + (x3 + 2 x)J1(x) · I0(x) + (x4 − 4 x2)J1(x) · I1(x)

]With∫

xn·J1(x)·I0(x) dx =12


holdsP6(x) = −x6 − 8 x4 + 24 x2 , Q6(x) = 5 x5 + 4 x3 − 24 x

R6(x) = x5 + 28 x3 − 24 x , S6(x) = x6 − 12 x4 − 32 x2

P8(x) = −x8 − 18 x6 + 192 x4 + 432 x2 , Q8(x) = 7 x7 + 6 x5 − 600 x3 − 432 x

R8(x) = x7 + 102 x5 − 168 x3 − 432 x , S8(x) = x8 − 24 x6 − 216 x4 + 768 x2

P10(x) = −x10 − 32 x8 + 720 x6 + 6144 x4 − 17280 x2

Q10(x) = 9 x9 + 8 x7 − 3696 x5 − 3648 x3 + 17280 x

R10(x) = x9 + 248 x7 − 624 x5 − 20928 x3 + 17280 x

S10(x) = x10 − 40 x8 − 768 x6 + 8640 x4 + 24576 x2

P12(x) = −x12 − 50 x10 + 1920 x8 + 36000 x6 − 368640 x4 − 864000 x2

Q12(x) = 11 x11 + 10 x9 − 13680 x7 − 15840 x5 + 1169280 x3 + 864000 x

R12(x) = x11 + 490 x9 − 1680 x7 − 200160 x5 + 305280 x3 + 864000 x

S12(x) = x12 − 60 x10 − 2000 x8 + 46080x6 + 432000 x4 − 1474560 x2

P14(x) = −x14 − 72 x12 + 4200 x10 + 138240 x8 − 3024000x6 − 26542080 x4 + 72576000 x2

Q14(x) = 13 x13 + 12 x11 − 38280 x9 − 48960 x7 + 15707520 x5 + 16796160 x3 − 72576000 x

R14(x) = x13 + 852 x11 − 3720 x9 − 1056960 x7 + 2436480x5 + 89372160 x3 − 72576000 x

S14(x) = x14 − 84 x12 − 4320 x10 + 168000 x8 + 3317760 x6 − 36288000 x4 − 106168320 x2

P16(x) =

= −x16 − 98 x14 + 8064 x12 + 411600 x10 − 15482880 x8 − 296352000x6 + 2972712960x4 + 7112448000x2

Q16(x) =

= 15 x15 + 14 x13 − 89544 x11 − 122640 x9 + 111323520 x7 + 145877760 x5 − 9501649920 x3 − 7112448000 x

R16(x) =

= x15 + 1358 x13 − 7224 x11 − 3993360x9 + 12539520 x7 + 1632234240 x5 − 2389201920 x3 − 7112448000 x

S16(x) =

= x16 − 112 x14 − 8232 x12 + 483840 x10 + 16464000 x8 − 371589120x6 − 3556224000x4 + 11890851840x2

78

XX. Integrals of the type∫

x2n+1 · J1(x) · I1(x) dx

∫x · J1(x) · I1(x) dx =

x

2[−J0(x) · I1(x) + J1(x) · I0(x)]∫

x3 · J1(x) · I1(x) dx =12[2 x2J0(x) · I0(x) + (−x3 − 2 x)J0(x) · I1(x) + (x3 − 2 x)J1(x) · I0(x)

]With∫

xn·J1(x)·I0(x) dx =12


holdsP5(x) = 4 x4 , Q5(x) = −x5 − 8 x3 , R5(x) = x5 − 8 x3 , S5(x) = 16 x2

P7(x) = 6 x6 − 144 x2 , Q7(x) = −x7 − 18 x5 + 72 x3 + 144 x

R7(x) = x7 − 18 x5 − 72 x3 + 144 x , S7(x) = 72 x4

P9(x) = 8 x8 − 1536 x4 , Q9(x) = −x9 − 32 x7 + 384 x5 + 3072 x3

R9(x) = x9 − 32 x7 − 384 x5 + 3072 x3 , S9(x) = 192 x6 − 6144 x2

P11(x) = 10 x10 − 7200 x6 + 172800 x2

Q11(x) = −x11 − 50 x9 + 1200 x7 + 21600 x5 − 86400 x3 − 172800 x

R11(x) = x11 − 50 x9 − 1200 x7 + 21600 x5 + 86400 x3 − 172800 x

S11(x) = 400 x8 − 86400 x4

P13(x) = 12 x12 − 23040 x8 + 4423680 x4

Q13(x) = −x13 − 72 x11 + 2880 x9 + 92160 x7 − 1105920x5 − 8847360 x3

R13(x) = x13 − 72 x11 − 2880 x9 + 92160 x7 + 1105920x5 − 8847360 x3

S13(x) = 720 x10 − 552960 x6 + 17694720 x2

P15(x) = 14 x14 − 58800 x10 + 42336000 x6 − 1016064000 x2

Q15(x) = −x15−98 x13 +5880 x11 +294000 x9−7056000x7−127008000 x5 +508032000x3 +1016064000x

R15(x) = x15 − 98 x13 − 5880 x11 + 294000x9 + 7056000x7 − 127008000 x5 − 508032000 x3 + 1016064000 x

S15(x) = 1176 x12 − 2352000 x8 + 508032000x4

79

XXI. Integrals of the type∫

x2n+1 · J0(αx) · I0(βx) dx

∫xJ0(αx) · I0(βx) dx =

β x

α2 + β2J0(αx) · I1(βx) +

α x

α2 + β2J1(αx) · I0(βx)∫

x3J0(αx) · I0(βx) dx =

= 2

(α2 − β2

)x2

(α2 + β2)2J0(αx) · I0(βx) +

(β x3

α2 + β2− 4

β(−β2 + α2

)x

(α2 + β2)3

)J0(αx) · I1(βx)+

+

(α x3

α2 + β2− 4

α(α2 − β2

)x

(α2 + β2)3

)J1(αx) · I0(βx)− 4

β α x2

(α2 + β2)2J1(αx) · I1(βx)

With ∫xn · J0(αx) · I0(βx) dx =

= Pn(x)J0(αx) · I0(βx) + Qn(x)J0(αx) · I1(βx) + Rn(x)J1(αx) · I0(βx) + Sn(x)J1(αx) · I1(βx)

holds

P5(x) = 4

(−β2 + α2

)x4

(α2 + β2)2− 32

(α4 − 4 α2β2 + β4

)x2

(α2 + β2)4

Q5(x) =β x5

α2 + β2− 16

β(2 α2 − β2

)x3

(α2 + β2)3+ 64

β(α4 − 4 α2β2 + β4

)x

(α2 + β2)5

R5(x) =α x5

α2 + β2− 16

α(α2 − 2 β2

)x3

(α2 + β2)3+ 64

α(α4 − 4 α2β2 + β4

)x

(α2 + β2)5

S5(x) = −8α β x4

(α2 + β2)2+ 96

α β(−β2 + α2

)x2

(α2 + β2)4

P7(x) = 6

(−β2 + α2

)x6

(α2 + β2)2− 48

(3 α4 − 14 α2β2 + 3 β4

)x4

(α2 + β2)4+ 1152

(α6 − 9 α4β2 + 9 α2β4 − β6

)x2

(α2 + β2)6

Q7(x) =β x7

α2 + β2− 12

β(7 α2 − 3 β2

)x5

(α2 + β2)3+ 192

β(8 α4 − 19 α2β2 + 3 β4

)x3

(α2 + β2)5−

−2304β(α6 − 9 α4β2 + 9 α2β4 − β6

)x

(α2 + β2)7

R7(x) =α x7

α2 + β2− 12

α(3 α2 − 7 β2

)x5

(α2 + β2)3+ 192

α(3 α4 − 19 α2β2 + 8 β4

)x3

(α2 + β2)5−

−2304α(α6 − 9 α4β2 + 9 α2β4 − β6

)x

(α2 + β2)7

S7(x) = −12β α x6

(α2 + β2)2+ 480

β α(−β2 + α2

)x4

(α2 + β2)4− 384

β α(11 α4 − 38 α2β2 + 11 β4

)x2

(α2 + β2)6

P9(x) = 8

(−β2 + α2

)x8

(α2 + β2)2− 384

(α4 − 5 α2β2 + β4

)x6

(α2 + β2)4+ 3072

(3 α6 − 32 α4β2 + 32 α2β4 − 3 β6

)x4

(α2 + β2)6−

−73728

(α8 − 16 β2α6 + 36 β4α4 − 16 β6α2 + β8

)x2

(α2 + β2)8

Q9(x) =β x9

α2 + β2− 32

β(5 α2 − 2 β2

)x7

(α2 + β2)3+ 768

β(10 α4 − 22 α2β2 + 3 β4

)x5

(α2 + β2)5−

80

−6144β(19 α6 − 108 α4β2 + 77 α2β4 − 6 β6

)x3

(α2 + β2)7+ 147456

β(α8 − 16 β2α6 + 36 β4α4 − 16 β6α2 + β8

)x

(α2 + β2)9

R9(x) =α x9

α2 + β2− 32

α(2 α2 − 5 β2

)x7

(α2 + β2)3+ 768

α(3 α4 − 22 α2β2 + 10 β4

)x5

(α2 + β2)5−

−6144α(6 α6 − 77 α4β2 + 108 α2β4 − 19 β6

)x3

(α2 + β2)7+ 147456

α(α8 − 16 β2α6 + 36 β4α4 − 16 β6α2 + β8

)x

(α2 + β2)9

S9(x) = −16α β x8

(α2 + β2)2+ 1344

α β(−β2 + α2

)x6

(α2 + β2)4− 3072

α β(13 α4 − 44 α2β2 + 13 β4

)x4

(α2 + β2)6+

+61440α β

(5 α6 − 37 α4β2 + 37 α2β4 − 5 β6

)x2

(α2 + β2)8

P11(x) = 10

(−β2 + α2

)x10

(α2 + β2)2−160

(5 α4 − 26 α2β2 + 5 β4

)x8

(α2 + β2)4+7680

(5 α6 − 58 α4β2 + 58 α2β4 − 5 β6

)x6

(α2 + β2)6−

−61440

(15 α8 − 283 β2α6 + 664 β4α4 − 283 β6α2 + 15 β8

)x4

(α2 + β2)8+

+7372800

(α10 − 25 α8β2 + 100 α6β4 − 100 α4β6 + 25 α2β8 − β10

)x2

(α2 + β2)10

Q11(x) =β x11

α2 + β2− 20

β(13 α2 − 5 β2

)x9

(α2 + β2)3+ 640

β(37 α4 − 79 α2β2 + 10 β4

)x7

(α2 + β2)5−

−15360β(62 α6 − 332 α4β2 + 221 α2β4 − 15 β6

)x5

(α2 + β2)7+

+122880β(107 α8 − 1119 β2α6 + 1881 β4α4 − 643 β6α2 + 30 β8

)x3

(α2 + β2)9−

−14745600β(α10 − 25 α8β2 + 100 α6β4 − 100 α4β6 + 25 α2β8 − β10

)x

(α2 + β2)11

R11(x) =α x11

α2 + β2− 20

α(5 α2 − 13 β2

)x9

(α2 + β2)3+ 640

α(10 α4 − 79 α2β2 + 37 β4

)x7

(α2 + β2)5−

−15360α(15 α6 − 221 α4β2 + 332 α2β4 − 62 β6

)x5

(α2 + β2)7+

+122880α(30 α8 − 643 β2α6 + 1881 β4α4 − 1119 β6α2 + 107 β8

)x3

(α2 + β2)9−

−14745600α(α10 − 25 α8β2 + 100 α6β4 − 100 α4β6 + 25 α2β8 − β10

)x

(α2 + β2)11

S11(x) = −20β α x10

(α2 + β2)2+ 2880

β α(−β2 + α2

)x8

(α2 + β2)4− 3840

β α(47 α4 − 158 α2β2 + 47 β4

)x6

(α2 + β2)6+

+430080β α

(11 α6 − 79 α4β2 + 79 α2β4 − 11 β6

)x4

(α2 + β2)8−

−245760β α

(137 α8 − 1762 β2α6 + 3762 β4α4 − 1762 β6α2 + 137 β8

)x2

(α2 + β2)10

81

XXII. Integrals of the type∫

x2n · J0(αx) · I1(βx) dx

∫x2J0(αx) · I1(βx) dx =

=β x2

α2 + β2J0(αx)·I0(βx)−2

β2x

(α2 + β2)2J0(αx)·I1(βx)−2

βαx

(α2 + β2)2J1(αx)·I0(βx)+

α x2

α2 + β2J1(αx)·I1(βx)

∫x4J0(αx) · I1(βx) dx =

(β x4

α2 + β2− 8

β(−β2 + 2 α2

)x2

(α2 + β2)3

)J0(αx) · I0(βx)+

+

(2

(α2 − 2 β2

)x3

(α2 + β2)2+ 16

β2(−β2 + 2 α2

)x

(α2 + β2)4

)J0(αx) · I1(βx)+

+

(−6

β α x3

(α2 + β2)2+ 16

β α(2 α2 − β2

)x

(α2 + β2)4

)J1(αx)·I0(βx)+

(α x4

α2 + β2− 4

α(α2 − 5 β2

)x2

(α2 + β2)3

)J1(αx)·I1(βx)



holds

P6(x) =β x6

α2 + β2− 8

β(7 α2 − 3 β2

)x4

(α2 + β2)3+ 192

β(3 α4 − 6 β2α2 + β4

)x2

(α2 + β2)5

Q6(x) = 2

(2 α2 − 3 β2

)x5

(α2 + β2)2− 32

(α4 − 11 β2α2 + 3 β4

)x3

(α2 + β2)4− 384

β2(3 α4 − 6 β2α2 + β4

)x

(α2 + β2)6

R6(x) = −10α β x5

(α2 + β2)2+ 32

α β(8 α2 − 7 β2

)x3

(α2 + β2)4− 384

α β(3 α4 − 6 β2α2 + β4

)x

(α2 + β2)6

S6(x) =α x6

α2 + β2− 16

α(α2 − 4 β2

)x4

(α2 + β2)3+ 64

α(α4 − 19 β2α2 + 10 β4

)x2

(α2 + β2)5

P8(x) =β x8

α2 + β2− 24

β(5 α2 − 2 β2

)x6

(α2 + β2)3+ 192

β(21 α4 − 43 β2α2 + 6 β4

)x4

(α2 + β2)5−

−9216β(4 α6 − 18 α4β2 + 12 α2β4 − β6

)x2

(α2 + β2)7

Q8(x) = 2

(3 α2 − 4 β2

)x7

(α2 + β2)2−48

(3 α4 − 26 β2α2 + 6 β4

)x5

(α2 + β2)4+384

(3 α6 − 86 α4β2 + 109 α2β4 − 12 β6

)x3

(α2 + β2)6+

+18432β2(4 α6 − 18 α4β2 + 12 α2β4 − β6

)x

(α2 + β2)8

R8(x) = −14α β x7

(α2 + β2)2+ 48

α β(18 α2 − 17 β2

)x5

(α2 + β2)4− 1920

α β(9 α4 − 26 β2α2 + 7 β4

)x3

(α2 + β2)6+

+18432α β

(4 α6 − 18 α4β2 + 12 α2β4 − β6

)x

(α2 + β2)8

S8(x) =α x8

α2 + β2− 12

α(3 α2 − 11 β2

)x6

(α2 + β2)3+ 192

α(3 α4 − 44 β2α2 + 23 β4

)x4

(α2 + β2)5−

−768α(3 α6 − 131 α4β2 + 239 α2β4 − 47 β6

)x2

(α2 + β2)7

82

P10(x) =β x10

α2 + β2− 16

β(13 α2 − 5 β2

)x8

(α2 + β2)3+ 768

β(19 α4 − 39 β2α2 + 5 β4

)x6

(α2 + β2)5−

−6144β(69 α6 − 332 α4β2 + 214 α2β4 − 15 β6

)x4

(α2 + β2)7+

+737280β(5 α8 − 40 α6β2 + 60 α4β4 − 20 α2β6 + β8

)x2

(α2 + β2)9

Q10(x) = 2

(4 α2 − 5 β2

)x9

(α2 + β2)2−64

(6 α4 − 47 β2α2 + 10 β4

)x7

(α2 + β2)4+1536

(6 α6 − 136 α4β2 + 158 α2β4 − 15 β6

)x5

(α2 + β2)6−

−12288

(6 α8 − 337 α6β2 + 1018 α4β4 − 499 α2β6 + 30 β8

)x3

(α2 + β2)8−

−1474560β2(5 α8 − 40 α6β2 + 60 α4β4 − 20 α2β6 + β8

)x

(α2 + β2)10

R10(x) = −18α β x9

(α2 + β2)2+ 64

α β(32 α2 − 31 β2

)x7

(α2 + β2)4− 10752

α β(9 α4 − 28 β2α2 + 8 β4

)x5

(α2 + β2)6+

+12288α β

(144 α6 − 863 α4β2 + 782 α2β4 − 101 β6

)x3

(α2 + β2)8−

−1474560α β

(5 α8 − 40 α6β2 + 60 α4β4 − 20 α2β6 + β8

)x

(α2 + β2)10

S10(x) =α x10

α2 + β2− 32

α(2 α2 − 7 β2

)x8

(α2 + β2)3+ 384

α(6 α4 − 79 β2α2 + 41 β4

)x6

(α2 + β2)5−

−6144α(6 α6 − 199 α4β2 + 354 α2β4 − 71 β6

)x4

(α2 + β2)7+

+24576α(6 α8 − 481 α6β2 + 1881 α4β4 − 1281 α2β6 + 131 β8

)x2

(α2 + β2)9

83

XXIII. Integrals of the type∫

x2n · J1(αx) · I0(βx) dx

∫x2J1(αx) · I0(βx) dx =

= − α x2

α2 + β2J0(αx)·I0(βx)+2

α xβ

(α2 + β2)2J0(αx)·I1(βx)+2

α2x

(α2 + β2)2J1(αx)·I0(βx)+

β x2

α2 + β2J1(αx)·I1(βx)

∫x4J1(αx) · I0(βx) dx =

(− α x4

α2 + β2+ 8

α(−2 β2 + α2

)x2

(α2 + β2)3

)J0(αx) · I0(βx) +

+

(6

α β x3

(α2 + β2)2− 16

α β(−2 β2 + α2

)x

(α2 + β2)4

)J0(αx) · I1(βx) +

+

(2

(2 α2 − β2

)x3

(α2 + β2)2− 16

α2(α2 − 2 β2

)x

(α2 + β2)4

)J1(αx)·I0(βx)+

(β x4

α2 + β2− 4

β(5 α2 − β2

)x2

(α2 + β2)3

)J1(αx)·I1(βx)



holds

P6(x) = − α x6

α2 + β2+ 8

α(3 α2 − 7 β2

)x4

(α2 + β2)3− 192

α(α4 − 6 α2β2 + 3 β4

)x2

(α2 + β2)5

Q6(x) = 10α β x5

(α2 + β2)2− 32

α β(7 α2 − 8 β2

)x3

(α2 + β2)4+ 384

α β(α4 − 6 α2β2 + 3 β4

)x

(α2 + β2)6

R6(x) = 2

(3 α2 − 2 β2

)x5

(α2 + β2)2− 32

(3 α4 − 11 α2β2 + β4

)x3

(α2 + β2)4+ 384

α2(α4 − 6 α2β2 + 3 β4

)x

(α2 + β2)6

S6(x) =β x6

α2 + β2− 16

β(4 α2 − β2

)x4

(α2 + β2)3+ 64

β(10 α4 − 19 α2β2 + β4

)x2

(α2 + β2)5

P8(x) = − α x8

α2 + β2+ 24

α(2 α2 − 5 β2

)x6

(α2 + β2)3− 192

α(6 α4 − 43 α2β2 + 21 β4

)x4

(α2 + β2)5+

+9216α(α6 − 12 α4β2 + 18 α2β4 − 4 β6

)x2

(α2 + β2)7

Q8(x) = 14α β x7

(α2 + β2)2− 48

α β(17 α2 − 18 β2

)x5

(α2 + β2)4+ 1920

α β(7 α4 − 26 α2β2 + 9 β4

)x3

(α2 + β2)6−

−18432α β

(α6 − 12 α4β2 + 18 α2β4 − 4 β6

)x

(α2 + β2)8

R8(x) = 2

(4 α2 − 3 β2

)x7

(α2 + β2)2−48

(6 α4 − 26 α2β2 + 3 β4

)x5

(α2 + β2)4+384

(12 α6 − 109 α4β2 + 86 α2β4 − 3 β6

)x3

(α2 + β2)6−

−18432α2(α6 − 12 α4β2 + 18 α2β4 − 4 β6

)x

(α2 + β2)8

S8(x) =β x8

α2 + β2− 12

β(11 α2 − 3 β2

)x6

(α2 + β2)3+ 192

β(23 α4 − 44 α2β2 + 3 β4

)x4

(α2 + β2)5−

−768β(47 α6 − 239 α4β2 + 131 α2β4 − 3 β6

)x2

(α2 + β2)7

84

P10(x) = − α x10

α2 + β2+ 16

α(5 α2 − 13 β2

)x8

(α2 + β2)3− 768

α(5 α4 − 39 α2β2 + 19 β4

)x6

(α2 + β2)5+

+6144α(15 α6 − 214 α4β2 + 332 α2β4 − 69 β6

)x4

(α2 + β2)7−

−737280α(α8 − 20 α6β2 + 60 β4α4 − 40 β6α2 + 5 β8

)x2

(α2 + β2)9

Q10(x) = 18α β x9

(α2 + β2)2− 64

α β(31 α2 − 32 β2

)x7

(α2 + β2)4+ 10752

α β(8 α4 − 28 α2β2 + 9 β4

)x5

(α2 + β2)6−

−12288α β

(101 α6 − 782 α4β2 + 863 α2β4 − 144 β6

)x3

(α2 + β2)8+

+1474560α β

(α8 − 20 α6β2 + 60 β4α4 − 40 β6α2 + 5 β8

)x

(α2 + β2)10

R10(x) = 2

(5 α2 − 4 β2

)x9

(α2 + β2)2− 64

(10 α4 − 47 α2β2 + 6 β4

)x7

(α2 + β2)4+

+1536

(15 α6 − 158 α4β2 + 136 α2β4 − 6 β6

)x5

(α2 + β2)6−

−12288

(30 α8 − 499 α6β2 + 1018 β4α4 − 337 β6α2 + 6 β8

)x3

(α2 + β2)8+

+1474560α2(α8 − 20 α6β2 + 60 β4α4 − 40 β6α2 + 5 β8

)x

(α2 + β2)10

S10(x) =β x10

α2 + β2− 32

β(7 α2 − 2 β2

)x8

(α2 + β2)3+ 384

β(41 α4 − 79 α2β2 + 6 β4

)x6

(α2 + β2)5−

− 6144β(71 α6 − 354 α4β2 + 199 α2β4 − 6 β6

)x4

(α2 + β2)7+

+24576β(131 α8 − 1281 α6β2 + 1881 β4α4 − 481 β6α2 + 6 β8

)x2

(α2 + β2)9

85

XXIV. Integrals of the type∫

x2n+1 · J1(αx) · I1(βx) dx

∫xJ1(αx) · I1(βx) dx = − α x

α2 + β2J0(αx) · I1(βx) +

β x

α2 + β2J1(αx) · I0(βx)∫

x3J1(αx) · I1(βx) dx = 4β α x2

(α2 + β2)2J0(αx) · I0(βx) +

+

(− α x3

α2 + β2− 8

β2α x

(α2 + β2)3

)J0(αx) · I1(βx) +

(β x3

α2 + β2− 8

α2β x

(α2 + β2)3

)J1(αx) · I0(βx) +

+2

(α2 − β2

)x2

(α2 + β2)2J1(αx) · I1(βx)



holds

P5(x) = 8α β x4

(α2 + β2)2− 96

α β(α2 − β2

)x2

(α2 + β2)4

Q5(x) = − α x5

α2 + β2+ 8

α(α2 − 5 β2

)x3

(α2 + β2)3+ 192

β2α(α2 − β2

)x

(α2 + β2)5

R5(x) =β x5

α2 + β2− 8

β(5 α2 − β2

)x3

(α2 + β2)3+ 192

α2β(α2 − β2

)x

(α2 + β2)5

S5(x) = 4

(α2 − β2

)x4

(α2 + β2)2− 16

(α4 − 10 β2α2 + β4

)x2

(α2 + β2)4

P7(x) = 12α β x6

(α2 + β2)2− 480

α β(α2 − β2

)x4

(α2 + β2)4+ 4608

α β(α4 − 3 β2α2 + β4

)x2

(α2 + β2)6

Q7(x) = − α x7

α2 + β2+ 24

α(α2 − 4 β2

)x5

(α2 + β2)3− 192

α(α4 − 18 β2α2 + 11 β4

)x3

(α2 + β2)5−

− 9216β2α

(α4 − 3 β2α2 + β4

)x

(α2 + β2)7

R7(x) =β x7

α2 + β2− 24

β(4 α2 − β2

)x5

(α2 + β2)3+ 192

β(11 α4 − 18 β2α2 + β4

)x3

(α2 + β2)5−

− 9216α2β

(α4 − 3 β2α2 + β4

)x

(α2 + β2)7

S7(x) = 6

(α2 − β2

)x6

(α2 + β2)2− 96

(α4 − 8 β2α2 + β4

)x4

(α2 + β2)4+ 384

(α6 − 29 α4β2 + 29 β4α2 − β6

)x2

(α2 + β2)6

P9(x) = 16α β x8

(α2 + β2)2− 1344

α β(α2 − β2

)x6

(α2 + β2)4+ 1536

α β(27 α4 − 86 β2α2 + 27 β4

)x4

(α2 + β2)6−

− 368640α β

(α6 − 6 α4β2 + 6 β4α2 − β6

)x2

(α2 + β2)8

Q9(x) = − α x9

α2 + β2+ 16

α(3 α2 − 11 β2

)x7

(α2 + β2)3− 384

α(3 α4 − 43 β2α2 + 24 β4

)x5

(α2 + β2)5+

86

+3072α(3 α6 − 121 α4β2 + 239 β4α2 − 57 β6

)x3

(α2 + β2)7+ 737280

α β2(α6 − 6 α4β2 + 6 β4α2 − β6

)x

(α2 + β2)9

R9(x) =β x9

α2 + β2− 16

β(11 α2 − 3 β2

)x7

(α2 + β2)3+ 384

β(24 α4 − 43 β2α2 + 3 β4

)x5

(α2 + β2)5−

− 3072β(57 α6 − 239 α4β2 + 121 β4α2 − 3 β6

)x3

(α2 + β2)7+ 737280

α2β(α6 − 6 α4β2 + 6 β4α2 − β6

)x

(α2 + β2)9

S9(x) = 8

(α2 − β2

)x8

(α2 + β2)2− 96

(3 α4 − 22 β2α2 + 3 β4

)x6

(α2 + β2)4+ 1536

(3 α6 − 67 α4β2 + 67 β4α2 − 3 β6

)x4

(α2 + β2)6−

− 6144

(3 α8 − 178 α6β2 + 478 α4β4 − 178 α2β6 + 3 β8

)x2

(α2 + β2)8

P11(x) = 20α β x10

(α2 + β2)2− 2880

α β(α2 − β2

)x8

(α2 + β2)4+ 46080

α β(4 α4 − 13 β2α2 + 4 β4

)x6

(α2 + β2)6−

− 2580480α β

(2 α6 − 13 α4β2 + 13 β4α2 − 2 β6

)x4

(α2 + β2)8+

+44236800α β

(α8 − 10 α6β2 + 20 α4β4 − 10 α2β6 + β8

)x2

(α2 + β2)10

Q11(x) = − α x11

α2 + β2+ 40

α(2 α2 − 7 β2

)x9

(α2 + β2)3− 3840

α(α4 − 13 β2α2 + 7 β4

)x7

(α2 + β2)5+

+92160α(α6 − 32 α4β2 + 59 β4α2 − 13 β6

)x5

(α2 + β2)7−

− 737280α(α8 − 73 α6β2 + 300 α4β4 − 227 α2β6 + 29 β8

)x3

(α2 + β2)9−

− 88473600β2α

(α8 − 10 α6β2 + 20 α4β4 − 10 α2β6 + β8

)x

(α2 + β2)11

R11(x) =β x11

α2 + β2− 40

β(7 α2 − 2 β2

)x9

(α2 + β2)3+ 3840

β(7 α4 − 13 β2α2 + β4

)x7

(α2 + β2)5−

− 92160β(13 α6 − 59 α4β2 + 32 β4α2 − β6

)x5

(α2 + β2)7+

+737280β(29 α8 − 227 α6β2 + 300 α4β4 − 73 α2β6 + β8

)x3

(α2 + β2)9−

− 88473600α2β

(α8 − 10 α6β2 + 20 α4β4 − 10 α2β6 + β8

)x

(α2 + β2)11

S11(x) = 10

(α2 − β2

)x10

(α2 + β2)2− 640

(α4 − 7 β2α2 + β4

)x8

(α2 + β2)4+ 23040

(α6 − 20 α4β2 + 20 β4α2 − β6

)x6

(α2 + β2)6−

− 368640

(α8 − 45 α6β2 + 118 α4β4 − 45 α2β6 + β8

)x4

(α2 + β2)8+

+1474560

(α10 − 102 α8β2 + 527 β4α6 − 527 β6α4 + 102 β8α2 − β10

)x2

(α2 + β2)10

87

XXV. Integrals of the type∫

x−mJn0 (x)J4−n

1 (x) dx∫J2

0 (x)J21 (x)

xdx = −x2 + 1

4J4

0 (x) +x

2J3

0 (x)J1(x)− x2 + 12

J20 (x)J2

1 (x) +x

2J0(x)J3

1 (x)− x2

4J4

1 (x)∫I20 (x)I2

1 (x)x

dx = −x2 − 14

I40 (x) +

x

2I30 (x)I1(x) +

x2 − 12

I20 (x)I2

1 (x)− x

2I0(x)I3

1 (x)− x2

4I41 (x)

∫J0(x)J3

1 (x) dx =14[x2J4

0 (x)− 2xJ30 (x)J1(x) + 2x2J2

0 (x)J21 (x)− 2xJ0(x)J3

1 (x) + x2J41 (x)

]∫

I0(x)I31 (x) dx =

14[−x2I4

0 (x) + 2xI30 (x)I1(x) + 2x2I2

0 (x)I21 (x)− 2xI0(x)I3

1 (x)− x2I41 (x)

]∫

J0(x)J31 (x)

x2dx = −4x2 + 3

16J4

0 (x)+x

2J3

0 (x)J1(x)−4x2 + 38

J20 (x)J2

1 (x)+2x2 − 1

4xJ0(x)J3

1 (x)−4x2 − 116

J41 (x)∫

I0(x)I31 (x)

x2dx = −4x2 − 3

16I40 (x)+

x

2I30 (x)I1(x)+

4x2 − 38

I20 (x)I2

1 (x)− 2x2 + 14x

I0(x)I31 (x)− 4x2 + 1

16I41 (x)

∫J4

1 (x)x

dx =14[x2J4

0 (x)− 2xJ30 (x)J1(x) + 2x2J2

0 (x)J21 (x)− 2xJ0(x)J3

1 (x) + (x2 − 1)J41 (x)

]∫

I41 (x)x

dx =14[−x2I4

0 (x) + 2xJ3I (x)I1(x) + 2x2I2

0 (x)I21 (x)− 2xI0(x)I3

1 (x)− (x2 + 1)I41 (x)

]∫

J41 (x)x3

dx = −4x2 + 324

J40 (x)+

x

3J3

0 (x)J1(x)−4x2 + 312

J20 (x)J2

1 (x)+2x2 − 1

6xJ0(x)J3

1 (x)−4x4 − x2 + 424x2

J41 (x)∫

I41 (x)x3

dx = −4x2 − 324

I40 (x)+

x

3I30 (x)I1(x)+

4x2 − 312

I20 (x)I2

1 (x)− 2x2 + 16x

I0(x)I31 (x)− 4x4 + x2 + 4

24x2I41 (x)

88

XXVI. Integrals of the type∫

xn+1/2 · Jν(x) dx

With the Lommel functions sµ,ν (see [7], 8.57, or [8], 10 -7) holds:∫ √xJ0(x) dx =

√xJ1(x)− x

4[2 s−1/2,1(x) J0(x) + s−3/2,0(x) J1(x)

],∫ √

xJ1(x) dx =x

2[s−1/2,0(x) J1(x)− 2 s1/2,0(x) J0(x)

].

x = t2 =⇒∫

x(2n−1)/2 Jν(x) dx = 2∫

t2n Jν(t2) dt

Differential equations: ∫ √xJ0(x) dx = y(x) =⇒ x2 y′′′ +

(x2 +

14

)y′ = 0∫ √

xJ1(x) dx = z(x) =⇒ x2 z′′′ +(

x2 − 34

)z′ = 0

Asymptotic expansions for x → +∞:

J0 :∫ x

0

√t J0(t) dt ∼ Γ2(3/4)

π+

√2π

∞∑k=0

ak

xksin(

x− 2k + 14

π

)Γ2(3/4)

π= 0.477 988 797 486 125

a0 = 1 , a1 =18

, a2 =25128

, a3 =4751024

, a4 =4927532768

, a5 =1636335262144

, a6 =1333080454194304

,

a7 =645675907533554432

, a8 =2905671971475

2147483648, a9 =

18638186048527517179869184

, . . .

k ak ak/ak−1 k ak ak/ak−1

0 1.000 000 000 - 5 6.242 122 650 4.15101 0.125 000 000 0.1250 6 31.783 114 67 5.09172 0.195 312 500 1.5625 7 192.426 415 5 6.05443 0.463 867 188 2.3750 8 1 353.058 951 7.03164 1.503 753 662 3.2418 9 10 848.852 14 8.0180

Let

D0,n(x) =Γ2(3/4)

π+

√2π

n∑k=0

ak

xksin(

x− 2k + 14

π

)−∫ x

0

√t J0(t) dt ,

then its first maximum and minimum values of interest are D0,n(x∗i,n).In the case x > x∗i,n holds |D0,n(x)| < |D0,n(x∗i,n)|.

n = 0, i = 1 2 3 4 5 6 7 8 9 10x∗i,0 1.143 4.058 7.146 10.264 13.394 16.527 19.664 22.801 25.940 29.079

103D0,0(x∗i ) 50.87 -21.784 13.293 -9.4707 7.3310 -5.9717 5.0342 -4.3496 3.8281 -3.4179

n = 1, i = 1 2 3 4 5 6 7 8 9 10x∗i,1 2.473 5.561 8.681 11.812 14.947 18.085 21.223 24.363 27.503 30.643

104D0,1(x∗i ) 158.930 -43.0361 19.1585 -10.6855 6.7779 -4.6705 3.4092 -2.5962 2.0420 -1.6478

n = 2, i = 3 4 5 6 7 8 9 10 11 12x∗i,2 7.100 10.233 13.369 16.508 19.647 22.787 25.927 29.068 32.209 35.350

105D0,2(x∗i ) -85.8809 31.2037 -14.5367 7.8808 -4.7310 3.0556 -2.0851 1.4850 -1.0945 0.8296

n = 3, i = 4 5 6 7 8 9 10 11 12 13x∗i,3 11.793 14.932 18.072 21.213 24.353 27.494 30.635 33.777 36.918 40.059

107D0,3(x∗i ) 548.133 -222.407 106.176 -56.7759 33.0053 -20.4558 13.3362 -9.0584 6.3648 -4.6012

89

n = 4, i = 5 6 7 8 9 10 11 12 13 14x∗i,4 16.499 19.649 22.780 25.922 29.063 32.204 35.345 38.487 41.628 44.770

108D0,4(x∗i ) -369.198 158.653 -76.8752 40.7761 -23.2093 13.9787 -8.8181 5.7815 -3.9164 2.7284

J1 :∫ x

0

√t J1(t) dt ∼ 4Γ2(5/4)

π−√

2π

∞∑k=0

bk

xksin(

x +2k + 1

4π

)4Γ2(5/4)

π= 1.046 049 620 053 102

b0 = 1 , b1 =38

, b2 = − 63128

, b3 =11131024

, b4 = −11157332768

, b5 =3643101262144

, b6 = −2942859154194304

,

b7 =14192615745

33554432, b8 = −6373074947085

2147483648, b9 =

40834492790206517179869184

, . . .

k bk |bk/bk−1| k bk |bk/bk−1|0 1.000 000 000 - 5 13.897 327 42 4.08151 0.375 000 000 0.3750 6 -70.163 229 70 5.04872 -0.492 187 500 1.3125 7 422.972 910 0 6.02843 1.086 914 063 2.2083 8 -2 967.694 284 7.01634 -3.404 937 744 3.1327 9 23 768.803 10 8.0092

Let

D1,n(x) =4Γ2(5/4)

π−√

2π

n∑k=0

bk

xksin(

x +2k + 1

4π

)−∫ x

0

√t J1(t) dt ,

then its first maximum and minimum values of interest are D1,n(x∗i,n).In the case x > x∗i,n holds |D1,n(x)| < |D1,n(x∗i,n)|.

n = 0, i = 1 2 3 4 5 6 7 8 9 10x∗i,0 2.470 6.470 8.675 11.807 14.943 18.081 21.220 24.360 27.500 30.641

103D1,0(x∗i ) -98.4511 50.5537 -33.4821 24.9168 -19.8073 16.4242 -14.0227 12.2312 -10.8443 9.7390

n = 1, i = 2 3 4 5 6 7 8 9 10 11x∗i,1 3.974 7.097 10.230 13.367 15.506 16.645 22.786 25.926 29.067 32.208

104D1,1(x∗i ) -194.456 70.4454 -35.5457 21.2582 -14.0945 10.0128 -7.4731 5.7878 -4.6133 3.7625

n = 2, i = 3 4 5 6 7 8 9 10 11 12x∗i,2 8.654 12.654 14.931 18.071 21.212 24.353 27.494 30.635 33.776 36.917

105D1,2(x∗i ) 117.286 -48.9485 24.7672 -14.1826 8.8517 -5.8853 4.1071 -2.9778 2.2269 -1.7083

n = 3, i = 4 5 6 7 8 9 10 11 12 13x∗i,3 13.358 16.498 19.639 22.780 29.921 29.063 32.204 35.345 38.487 41.628

107D0,3(x∗i ) -773.343 342.802 -173.926 97.2278 -58.4641 37.2099 -24.7842 17.1337 -12.2177 8.9436

n = 4, i = 4 5 6 7 8 9 10 11 12 13x∗i,4 11.785 14.926 18.067 21.208 24.349 27.491 30.632 33.774 36.915 40.057

107D1,4(x∗i ) 409.325 -133.167 53.0102 -24.2927 12.3500 -6.7990 3.9863 -2.4597 1.5831 -1.0557

90

Integrals: ∫x3/2 J0(x) dx = x3/2J1(x)− 1

2

∫ √xJ1(x) dx

∫x3/2 J1(x) dx = −x3/2J0(x) +

32

∫ √xJ0(x) dx

∫x5/2 J0(x) dx =

√x

[3x

2J0(x) + x2J1(x)

]− 9

4

∫ √xJ0(x) dx

∫x5/2 J1(x) dx =

√x

[−x2J0(x) +

52xJ1(x)

]− 5

4

∫ √xJ1(x) dx

∫x7/2 J0(x) dx =

√x

[5x2

2J0(x) +

(x3 − 25

4x

)J1(x)

]+

258

∫ √xJ1(x) dx

∫x7/2 J1(x) dx =

√x

[(−x3 +

214

x

)J0(x) +

72

x2J1(x)]− 63

8

∫ √xJ0(x) dx

∫x9/2 J0(x) dx =

√x

[(72

x3 − 1478

x

)J0(x) +

(x4 − 49

4x2

)J1(x)

]+

44116

∫ √xJ0(x) dx

∫x9/2 J1(x) dx =

√x

[(−x4 +

454

x2

)J0(x) +

(92

x3 − 2258

x

)J1(x)

]+

22516

∫ √xJ1(x) dx

∫x11/2 J0(x) dx =

√x

[(92

x4 − 4058

x2

)J0(x) +

(x5 − 81

4x3 +

202516

x

)J1(x)

]− 2025

32

∫ √xJ1(x) dx

∫x11/2 J1(x) dx =

√x

[(−x5 +

774

x3 − 161716

x

)J0(x) +

(112

x4 − 5398

x2

)J1(x)

]+

485132

∫ √xJ0(x) dx

∫x13/2 J0(x) dx =

√x

[(112

x5 − 8478

x3 +17787

32x

)J0(x) +

(x6 − 121

4x4 +

592916

x2

)J1(x)

]−

−5336164

∫ √xJ0(x) dx

∫x13/2 J1(x) dx =

√x

[(−x6 +

1174

x4 − 526516

x2

)J0(x) +

(132

x5 − 10538

x3 +26325

32x

)J1(x)

]−

−2632564

∫ √xJ1(x) dx

∫x15/2 J0(x) dx =

√x

[(132

x6 − 15218

x4 +68445

32x2

)J0(x)+

+(

x7 − 1694

x5 +13689

16x3 − 342225

64x +

342225128

)J1(x)

]+

342225128

∫ √xJ1(x) dx

∫x15/2 J1(x) dx =

√x

[(−x7 +

1654

x5 − 1270516

x3 +266805

64x

)J0(x)+

+(

152

x6 − 18158

x4 +88935

32x2

)J1(x)

]− 800415

128

∫ √xJ0(x) dx

91

∫x17/2 J0(x) dx =

√x

[(152

x7 − 24758

x5 +190575

32x3 − 4002075

128x

)J0(x)+

+(

x8 − 2254

x6 +27225

16x4 − 1334025

64x2

)J1(x)

]+

12006225256

∫ √xJ0(x) dx

∫x17/2 J1(x) dx =

√x

[(−x8 +

2214

x6 − 2585716

x4 +1163565

64x2

)J0(x)+

+(

17/2 x7 − 28738

x5 +232713

32x3 − 5817825

128x

)J1(x)

]+

5817825256

∫ √xJ1(x) dx

∫x19/2 J0(x) dx =

√x

[(172

x8 − 37578

x6 +439569

32x4 − 19780605

128x2

)J0(x)+(

x9 − 2894

x7 +48841

16x5 − 3956121

64x3 +

98903025256

x

)J1(x)

]− 98903025

512

∫ √xJ1(x) dx

∫x19/2 J1(x) dx =

√x

[(−x9 +

2854

x7 − 4702516

x5 +3620925

64x3 − 76039425

256x

)J0(x)+

+(

192

x8 − 42758

x6 +517275

32x4 − 25346475

128x2

)J1(x)

]+

228118275512

∫ √xJ0(x) dx

∫x21/2 J0(x) dx =

√x

[(192

x9 − 54158

x7 +893475

32x5 − 68797575

128x3 +

1444749075512

x

)J0(x)+

+(

x10 − 3614

x8 +81225

16x6 − 9828225

64x4 +

481583025256

x2

)J1(x)

]−

−43342472251024

∫ √xJ0(x) dx

∫x21/2 J1(x) dx =

√x

[(−x10 +

3574

x8 − 7889716

x6 +9230949

64x4 − 415392705

256x2

)J0(x)+

+(

21/2 x9 − 60698

x7 +1025661

32x5 − 83078541

128x3 +

2076963525512

x

)J1(x)

]− 2076963525

1024

∫ √xJ1(x) dx

∫x23/2 J0(x) dx =

√x

[(212

x10 − 74978

x8 +1656837

32x6 − 193849929

128x4 +

8723246805512

x2

)J0(x)+

(x11 − 441

4x9 +

12744916

x7 − 2153888164

x5 +1744649361

256x3 − 43616234025

1024x

)J1(x)

]+

+43616234025

2048

∫ √xJ1(x) dx∫

x23/2 J1(x) dx =

=√

x

[(−x11 +

4374

x9 − 12454516

x7 +20549925

64x5 − 1582344225

256x3 +

332292287251024

x

)J0(x)+

+(

23/2 x10 − 83038

x8 +1868175

32x6 − 226049175

128x4 +

11076409575512

x2

)J1(x)

]−

−996876861752048

∫ √xJ0(x) dx

92

∫x25/2 J0(x) dx =

=√

x

[(232

x11 − 100518

x9 +2864535

32x7 − 472648275

128x5 +

36393917175512

x3 − 7642722606752048

x

)J0(x)+

+(

x12 − 5294

x10 +190969

16x8 − 42968025

64x6 +

5199131025256

x4 − 2547574202251024

x2

)J1(x)

]+

+2292816782025

4096

∫ √xJ0(x) dx

∫x25/2 J1(x) dx =

=√

x

[(−x12 +

5254

x10 − 18742516

x8 +41420925

64x6 − 4846248225

256x4 +

2180811701251024

x2

)J0(x)+

+(

252

x11 − 110258

x9 +3186225

32x7 − 538472025

128x5 +

43616234025512

x3 − 10904058506252048

x

)J1(x)

]+

+1090405850625

4096

∫ √xJ1(x) dx

∫x27/2 J0(x) dx =

=√

x

[(252

x12 − 131258

x10 +4685625

32x8 − 1035523125

128x6 +

121156205625512

x4 − 54520292531252048

x2

)J0(x)+

+(

x13 − 6254

x11 +275625

16x9 − 79655625

64x7 +

13461800625256

x5 − 10904058506251024

x3+

272601462656254096

x

)J1(x)

]− 27260146265625

8192

∫ √xJ1(x) dx

∫x27/2 J1(x) dx =

√x

[(−x13 +

6214

x11 − 27137716

x9 +77342445

64x7 − 12761503425

256x5+

9826357637251024

x3 − 206353510382254096

x

)J0(x) +

(272

x12 − 142838

x10 +5156163

32x8 − 1160136675

128x6+

+140376537675

512x4 − 6878450346075

2048x2

)J1(x)

]+

619060531146758192

∫ √xJ0(x) dx

∫x29/2 J0(x) dx =

√x

[(272

x13 − 167678

x11 +7327179

32x9 − 2088246015

128x7 +

344560592475512

x5−

−265311656205752048

x3 +557154478032075

8192x

)J0(x) +

(x14 − 729

4x12 +

38564116

x10 − 13921640164

x8+

+31323690225

256x6 − 3790166517225

1024x4 +

1857181593440254096

x2

)J1(x)

]−1671463434096225

16384

∫ √xJ0(x) dx

∫x29/2 J1(x) dx =

√x

[(−x14 +

7254

x12 − 38062516

x10 +135883125

64x8 − 30030170625

256x6+

+3513529963125

1024x4 − 158108848340625

4096x2

)J0(x)+

(292

x13 − 181258

x11 +7993125

32x9 − 2310013125

128x7+

+390392218125

512x5 − 31621769668125

2048x3 +

7905442417031258192

x

)J1(x)

]−790544241703125

16384

∫ √xJ1(x) dx

93

∫J0(x)√

xdx = 2

√xJ0(x) + 2

∫ √xJ1(x) dx

∫J1(x)√

xdx = −2

√xJ1(x) + 2

∫ √xJ0(x) dx

∫x−3/2 · J0(x) dx =

√x

x[−2J0(x) + 4xJ1(x)]− 4

∫ √xJ0(x) dx

∫x−3/2 · J1(x) dx =

√x

3x[4xJ0(x)− 2J1(x)] +

43

∫ √xJ1(x) dx

∫x−5/2 · J0(x) dx =

√x

9x2

[(−8x2 − 6

)J0(x) + 4xJ1(x)

]− 8

9

∫ √xJ1(x) dx

∫x−5/2 · J1(x) dx =

√x

5x2[−4xJ0(x) + (8x2 − 2)J1(x)]− 8

5

∫ √xJ0(x) dx

∫x−7/2 · J0(x) dx =

√x

25 x3[(8x2 − 10)J0(x) + (−16 x3 + 4 x)J1(x)] +

1625

∫ √xJ0(x) dx

∫x−7/2 · J1(x) dx =

√x

63 x3[(−16 x3 − 12 x)J0(x) + (8 x2 − 18)J1(x)]− 16

63

∫ √xJ1(x) dx

∫x−9/2 · J0(x) dx =

√x

441 x4

[(32 x4 + 24 x2 − 126

)J0(x) + (−16 x3 + 36 x)J1(x)

]+

32441

∫ √xJ1(x) dx

∫x−9/2 · J1(x) dx =

√x

225 x4[(16x3 − 20 x)J0(x) + (−32 x4 + 8 x2 − 50)J1(x)] +

32225

∫ √xJ0(x) dx

∫x−11/2·J0(x) dx =

√x

2025 x5[(−32 x4+40 x2−450)J0(x)+(64 x5−16 x3+100 x)J1(x)]− 64

2025

∫ √xJ0(x) dx

∫x−11/2·J1(x) dx =

√x

4851 x5[(64x5+48 x3−252 x)J0(x)+(−32 x4+72 x2−882)J1(x)]+

644851

∫ √xJ1(x) dx

∫x−13/2 · J0(x) dx =

=√

x

53361 x6

[(−128 x6 − 96 x4 + 504 x2 − 9702)J0(x) + (64 x5 − 144 x3 + 1764 x)J1(x)

]− 128

53361

∫ √xJ1(x) dx

∫x−13/2 · J1(x) dx =

√x

26325 x6[(−64 x5 + 80 x3 − 900 x)J0(x)+

+(128x6 − 32 x4 + 200 x2 − 4050)J1(x)]− 12826325

∫ √xJ0(x) dx

∫x−15/2 · J0(x) dx =

√x

342225 x7[(128x6 − 160 x4 + 1800 x2 − 52650)J0(x)+

+(−256 x7 + 64 x5 − 400 x3 + 8100 x)J1(x)] +256

342225

∫ √xJ0(x) dx

∫x−15/2 · J1(x) dx =

√x

800415 x7[(−256 x7 − 192 x5 + 1008 x3 − 19404 x)J0(x)+

94

+(128x6 − 288 x4 + 3528 x2 − 106722)J1(x)]− 256800415

∫ √xJ1(x) dx

∫x−17/2 · J0(x) dx =

√x

12006225 x8[(512x8 + 384 x6 − 2016 x4 + 38808x2 − 1600830)J0(x)+

+(−256 x7 + 576 x5 − 7056 x3 + 213444 x)J1(x)] +512

12006225

∫ √xJ1(x) dx

∫x−17/2 · J1(x) dx =

√x

5817825 x8[(256x7 − 320 x5 + 3600 x3 − 105300 x)J0(x)+

+(−512 x8 + 128 x6 − 800 x4 + 16200 x2 − 684450)J1(x)] +512

5817825

∫ √xJ0(x) dx

∫x−19/2 · J0(x) dx =

√x

98903025 x9[(−512 x8 + 640 x6 − 7200 x4 + 210600 x2 − 11635650)J0(x)+

+(1024x9 − 256 x7 + 1600 x5 − 32400 x3 + 1368900x)J1(x)]− 102498903025

∫ √xJ0(x) dx

∫x−19/2 · J1(x) dx =

√x

228118275 x9[(1024x9 + 768 x7 − 4032 x5 + 77616x3 − 3201660 x)J0(x)+

+(−512 x8 + 1152 x6 − 14112 x4 + 426888 x2 − 24012450)J1(x)]∫ √

xJ1(x) dx∫x−21/2 · J0(x) dx =

=√

x

4334247225x10[(−2048 x10 − 1536 x8 + 8064 x6 − 155232 x4 + 6403320 x2 − 456236550)J0(x)+

+(1024x9− 2304 x7 +28224 x5− 853776 x3 +48024900 x)J1(x)]− 20484334247225

+1024

228118275

∫ √xJ1(x) dx

∫x−21/2 · J1(x) dx =

√x

2076963525 x10[(−1024 x9 + 1280 x7 − 14400 x5 + 421200 x3 − 23271300 x)J0(x)+

+(2048x10 − 512 x8 + 3200 x6 − 64800 x4 + 2737800x2 − 197806050)J1(x)]− 20482076963525

∫ √xJ0(x) dx∫

x−23/2 · J0(x) dx =

=√

x

43616234025x11[(2048x10 − 2560 x8 + 28800 x6 − 842400 x4 + 46542600 x2 − 4153927050)J0(x)+

+(−4096 x11+1024 x9−6400 x7+129600 x5−5475600 x3+395612100 x)J1(x)]+4096

43616234025

∫ √xJ0(x) dx∫

x−23/2 · J1(x) dx =

=√

x

99687686175x11[(−4096 x11 − 3072 x9 + 16128x7 − 310464 x5 + 12806640x3 − 912473100 x)J0(x)+

+(2048x10−4608 x8+56448x6−1707552 x4+96049800x2−8668494450)J1(x)]− 409699687686175

∫ √xJ1(x) dx

∫x−25/2 · J0(x) dx =

√x

2292816782025x12[(8192x12 + 6144 x10 − 32256 x8 + 620928 x6 − 25613280 x4+

95

+1824946200x2 − 199375372350)J0(x) + (−4096 x11 + 9216 x9 − 112896 x7 + 3415104 x5 − 192099600 x3+

+17336988900x)J1(x)] +8192

2292816782025

∫ √xJ1(x) dx

∫x−25/2 · J1(x) dx =

=√

x

1090405850625x12[(4096x11 − 5120 x9 + 57600 x7 − 1684800 x5 + 93085200x3 − 8307854100 x)J0(x)+

+(−8192 x12 + 2048 x10 − 12800 x8 + 259200 x6 − 10951200 x4 + 791224200x2 − 87232468050)J1(x)]+

+8192

1090405850625

∫ √xJ0(x) dx

96

XXVII. Integrals of the type∫

xn e±x Iν(x) dx

See also [1], 11.3.∫ex I0(x) dx = xex [I0(x)− I1(x)] ,

∫ex · I1(x) dx

x= ex [I0(x)− I1(x)]∫

ex I1(x) dx = ex [(1− x)I0(x) + xI1(x)]∫xex I0(x) dx =

xex

3[xI0(x) + (1− x)I1(x)]∫

xex I1(x) dx =xex

3[−xI0(x) + (2 + x)I1(x)]∫

x2ex I0(x) dx =xex

15[(2 x + 3 x2)I0(x) + (−4 + 4 x− 3 x2)I1(x)

]∫

x2ex I1(x) dx =xex

5[(x− x2)I0(x) + (−2 + 2x + x2)I1(x)

]∫

x3ex I0(x) dx =xex

35[(−6 x + 6 x2 + 5 x3)I0(x) + (12− 12 x + 9 x2 − 5 x3)I1(x)

]∫

x3ex I1(x) dx =xex

35[(−8 x + 8 x2 − 5 x3)I0(x) + (16− 16 x + 12 x2 + 5 x3)I1(x)

]∫

x4ex I0(x) dx =xex

315[(96 x− 96 x2 + 60 x3 + 35 x4)I0(x) + (−192 + 192 x− 144 x2 + 80 x3 − 35 x4)I1(x)

]∫

x4ex I1(x) dx =xex

63[(24 x− 24 x2 + 15 x3 − 7 x4)I0(x) + (−48 + 48 x− 36 x2 + 20 x3 + 7 x4)I1(x)

]∫

x5ex I0(x) dx =xex

693[(−480 x + 480 x2 − 300 x3 + 140 x4 + 63 x5)I0(x)+

+(960− 960 x + 720 x2 − 400 x3 + 175 x4 − 63 x5)I1(x)]∫

x5ex I1(x) dx =xex

231[(−192 x + 192 x2 − 120 x3 + 56 x4 − 21 x5)I0(x)+

+(384− 384 x + 288 x2 − 160 x3 + 70 x4 + 21 x5)I1(x)]∫

x6ex I0(x) dx =xex

1001[(1920x− 1920 x2 + 1200 x3 − 560 x4 + 210x5 + 77x6)I0(x)+

+ (−3840 + 3840 x− 2880 x2 + 1600 x3 − 700 x4 + 252x5 − 77x6)I1(x)]∫

x6ex I1(x) dx =xex

429[(960 x− 960 x2 + 600 x3 − 280 x4 + 105 x5 − 33 x6)I0(x)+

+(−1920 + 1920 x− 1440 x2 + 800 x3 − 350 x4 + 126 x5 + 33 x6)I1(x)]∫

x7ex I0(x) dx =xex

2145[(−13440 x + 13440 x2 − 8400 x3 + 3920 x4 − 1470 x5 + 462 x6 + 143 x7)I0(x)+

+(26880− 26880 x + 20160x2 − 11200 x3 + 4900 x4 − 1764 x5 + 539 x6 − 143 x7)I1(x)]∫

x7ex I1(x) dx =xex

2145[(−15360 x + 15360 x2 − 9600 x3 + 4480 x4 − 1680 x5 + 528 x6 − 143 x7)I0(x)+

+(30720− 30720 x + 23040x2 − 12800 x3 + 5600 x4 − 2016 x5 + 616 x6 + 143 x7)I1(x)]

Recurrence formulas:∫xnexI0(x) dx =

xnex

2n + 1[(n + x)I0(x)− xI1(x)]− n2

n + 1

∫xn−1exI0(x) dx∫

xnexI1(x) dx =xnex

2n + 1[(n + 1− x)I0(x) + xI1(x)] +

n(n + 1)n + 1

∫xn−1exI0(x) dx

The last formula refers to I0(x) instead of I1(x).

97

∫e−x I0(x) dx = xe−x [I0(x) + I1(x)] ,

∫e−x · I1(x) dx

x= e−x [I0(x) + I1(x)]∫

e−x I1(x) dx = e−x [(1 + x)I0(x) + xI1(x)]∫xe−x I0(x) dx =

xe−x

3[xI0(x) + (1 + x)I1(x)]∫

xe−x I1(x) dx =xe−x

3[xI0(x) + (−2 + x)I1(x)]∫

x2e−x I0(x) dx =xe−x

15[(−2 x + 3 x2)I0(x) + (4 + 4 x + 3 x2)I1(x)

]∫


5[(x + x2)I0(x) + (−2− 2x + x2)I1(x)

]∫


35[(−6 x− 6 x2 + 5 x3)I0(x) + (12 + 12 x + 9 x2 + 5 x3)I1(x)

]∫


35[(8 x + 8 x2 + 5 x3)I0(x) + (−16− 16 x− 12 x2 + 5 x3)I1(x)

]∫


315[(−96 x− 96 x2 − 60 x3 + 35 x4)I0(x) + (192 + 192 x + 144 x2 + 80 x3 + 35 x4)I1(x)

]∫


63[(24x + 24 x2 + 15 x3 + 7 x4)I0(x) + (−48− 48 x− 36 x2 − 20 x3 + 7 x4)I1(x)

]∫


693[(−480 x− 480 x2 − 300 x3 − 140 x4 + 63 x5)I0(x)+

+(960 + 960x + 720 x2 + 400 x3 + 175 x4 + 63 x5)I1(x)]

∫x5e−x I1(x) dx =

xe−x

231[(192 x + 192 x2 + 120 x3 + 56 x4 + 21 x5)I0(x)+

+(−384− 384 x− 288 x2 − 160 x3 − 70 x4 + 21 x5)I1(x)]


xe−x

1001[(−1920 x− 1920 x2 − 1200 x3 − 560 x4 − 210x5 + 77x6)I0(x)+

+ (3840 + 3840 x + 2880 x2 + 1600 x3 + 700 x4 + 252x5 + 77x6)I1(x)]


xe−x

429[(960 x + 960 x2 + 600 x3 + 280 x4 + 105 x5 + 33 x6)I0(x)+

+(−1920− 1920 x− 1440 x2 − 800 x3 − 350 x4 − 126 x5 + 33 x6)I1(x)]


xe−x

2145[(−13440 x− 13440 x2 − 8400 x3 − 3920 x4 − 1470 x5 − 462 x6 + 143 x7)I0(x)+

+(26880 + 26880x + 20160x2 + 11200x3 + 4900 x4 + 1764 x5 + 539 x6 + 143 x7)I1(x)]


xe−x

2145[(15360x + 15360x2 + 9600 x3 + 4480 x4 + 1680 x5 + 528 x6 + 143 x7)I0(x)+

+(−30720− 30720 x− 23040 x2 − 12800 x3 − 5600 x4 − 2016 x5 − 616 x6 + 143 x7)I1(x)]

Recurrence formulas:∫xne−xI0(x) dx =

xne−x

2n + 1[(x− n)I0(x) + xI1(x)]− n2

n + 1

∫xn−1e−xI0(x) dx

∫xne−xI1(x) dx =

xne−x

2n + 1[(n + 1 + x)I0(x) + xI1(x)]− n(n + 1)

n + 1

∫xn−1e−xI0(x) dx

The last formula refers to I0(x) instead of I1(x).

98

XXVIII. Integrals of the type∫

xn ·

sincos

x · Jν(x) dx

See also [1], 11.3. ∫sinx · J0(x) dx = x[sinx · J0(x)− cos x · J1(x)]∫cos x · J0(x) dx = x[cos x · J0(x) + sinx · J1(x)]∫

sinx · J1(x) dx = (x cos x− sinx)J0(x) + x sinx · J1(x)∫cos x · J1(x) dx = −(x sinx + cos x)J0(x) + x cos x · J1(x)∫

x sinx · J0(x) dx =x2

3sinx · J0(x) +

x sinx− x2 cos x

3· J1(x)∫

x cos x · J0(x) dx =x2

3cos x · J0(x) +

x2 sinx− x cos x

3· J1(x)∫

x sinx · J1(x) dx =x2

3· cos x · J0(x) +

x2 sinx− 2x cos x

3· J1(x)∫

x cos x · J1(x) dx = −x2

3sinx · J0(x) +

2x sinx + x2 cos x

3· J1(x)∫

x2 sinx · J0(x) dx =115[

3 x3 sinx− 2 x2 cos x]· J0(x) +

[4 x2 sinx + (4 x− 3 x3) cos x

]· J1(x)

∫

x2 cos x · J0(x) dx =115[

2 x2 sinx + 3 x3 cos x]· J0(x) +

[(−4 x + 3 x3) sinx + 4 x2 cos x

]· J1(x)

∫

x2 sinx · J1(x) dx =15[−x2 sinx + x3 cos x

]· J0(x) +

[(2 x + x3) sinx− 2 x2 cos x

]· J1(x)

∫

x2 cos x · J1(x) dx =15[−x3 sinx− x2 cos x

]· J0(x) +

[2 x2 sinx + (2 x + x3) cos x

]· J1(x)

∫

x3 sinx · J0(x) dx =135[

(6 x2 + 5 x4) sinx− 6 x3 cos x]· J0(x)+

+[(−12 x + 9 x3) sinx + (12x2 − 5 x4) cos x

]· J1(x)

∫x3 cos x · J0(x) dx =

135[

6 x3 sinx + (6 x2 + 5 x4) cos x]· J0(x)+

+[(−12 x2 + 5 x4) sinx + (−12 x + 9 x3) cos x

]· J1(x)

∫x3 sinx · J1(x) dx =

135[−8x3 sinx + (−8 x2 + 5 x4) cos x

]· J0(x)+

+[(16 x2 + 5 x4) sinx + (16x− 12 x3) cos x

]· J1(x)


135[

(8 x2 − 5 x4) sinx− 8 x3 cos x]· J0(x)+

+[(−16 x + 12 x3) sinx + (16x2 + 5 x4) cos x

]· J1(x)


1315

[(96x3 + 35 x5) sinx + (96x2 − 60 x4) cos x

]· J0(x)+

+[(−192 x2 + 80 x4) sinx + (−192 x + 144 x3 − 35 x5) cos x

]· J1(x)


1315

[(−96 x2 + 60 x4) sinx + (96x3 + 35 x5) cos x

]· J0(x)+

+[(192x− 144 x3 + 35 x5) sinx + (−192 x2 + 80 x4) cos x

]· J1(x)


1315

[(120x2 − 75x4) sinx + (−120 x3 + 35 x5) cos x

]· J0(x)+

99

+[(−240 x + 180 x3 + 35 x5) sinx + (240x2 − 100 x4) cos x

]· J1(x)


1315

[(120 x3 − 35 x5) sinx + (120x2 − 75 x4) cos x

]· J0(x)+

+[(−240 x2 + 100 x4) sinx + (−240 x + 180 x3 + 35 x5) cos x

]· J1(x)


1693

[(−480 x2 + 300 x4 + 63 x6) sinx + (480x3 − 140 x5) cos x

]· J0(x)+

+[(960 x− 720 x3 + 175 x5) sinx + (−960 x2 + 400 x4 − 63 x6) cos x

]· J1(x)


1693

[(−480 x3 + 140 x5) sinx + (−480 x2 + 300 x4 + 63 x6) cos x

]· J0(x)+

+[(960 x2 − 400 x4 + 63 x6) sinx + (960x− 720 x3 + 175 x5) cos x

]· J1(x)


1231

[(192x3 − 56x5) sinx + (192x2 − 120 x4 + 21 x6) cos x

]· J0(x)+

+[(−384 x2 + 160 x4 + 21 x6) sinx + (−384 x + 288 x3 − 70 x5) cos x

]· J1(x)


1231

[(−192 x2 + 120 x4 − 21 x6) sinx + (192x3 − 56 x5) cos x

]· J0(x)+

+[(384x− 288 x3 + 70 x5) sinx + (−384 x2 + 160 x4 + 21 x6) cos x

]· J1(x)


=1

1001[

(−1920 x3 + 560 x5 + 77 x7) sinx + (−1920 x2 + 1200 x4 − 210 x6) cos x]· J0(x)+

+[(3840x2 − 1600 x4 + 252 x6) sinx + (3840x− 2880 x3 + 700 x5 − 77 x7) cos x

]· J1(x)


=1

1001[

(1920x2 − 1200 x4 + 210 x6) sinx + (−1920 x3 + 560 x5 + 77 x7) cos x]· J0(x)+

+[(−3840 x + 2880 x3 − 700 x5 + 77 x7) sinx + (3840x2 − 1600 x4 + 252 x6) cos x

]· J1(x)


=1

429[

(−960x2 + 600x4 − 105x6) sinx + (960x3 − 280 x5 + 33 x7) cos x]· J0(x)+

+[(1920x− 1440 x3 + 350 x5 + 33 x7) sinx + (−1920 x2 + 800 x4 − 126 x6) cos x

]· J1(x)


=1

429[

(−960 x3 + 280 x5 − 33 x7) sinx + (−960 x2 + 600 x4 − 105 x6) cos x]· J0(x)+

+[(1920x2 − 800 x4 + 126 x6) sinx + (1920x− 1440 x3 + 350 x5 + 33 x7) cos x

]· J1(x)


=1

2145[

(13440 x2 − 8400 x4 + 1470 x6 + 143 x8) sinx + (−13440 x3 + 3920 x5 − 462 x7) cos x]· J0(x)+

+[(−26880 x + 20160 x3 − 4900 x5 + 539 x7) sinx + (26880x2 − 11200 x4 + 1764 x6 − 143 x8) cos x

]· J1(x)


=1

2145[

(13440x3 − 3920 x5 + 462 x7) sinx + (13440x2 − 8400 x4 + 1470 x6 + 143 x8) cos x]· J0(x)+

+[(−26880 x2 + 11200 x4 − 1764 x6 + 143 x8) sinx + (−26880 x + 20160 x3 − 4900 x5 + 539 x7) cos x

]· J1(x)

100


=1

2145[

(−15360x3 + 4480x5 − 528x7) sinx + (−15360 x2 + 9600 x4 − 1680 x6 + 143 x8) cos x]· J0(x)+

+[(30720x2 − 12800 x4 + 2016 x6 + 143 x8) sinx + (30720x− 23040 x3 + 5600 x5 − 616 x7) cos x

]· J1(x)


=1

2145[

(15360 x2 − 9600 x4 + 1680 x6 − 143 x8) sinx + (−15360 x3 + 4480 x5 − 528 x7) cos x]· J0(x)+

+[(−30720 x + 23040 x3 − 5600 x5 + 616 x7) sinx + (30720x2 − 12800 x4 + 2016 x6 + 143 x8) cos x

]· J1(x)

Recurrence formulas:Let

S(ν)n =

∫xn sinx · Jν(x) dx , C(ν)

n =∫

xn cos x · Jν(x) dx

andσ(ν)

n = xn sinx · Jν(x) , γ(ν)n = xn cos x · Jν(x) ,

then holds

S(0)n =

n2C(0)n−1 − nγ

(0)n + σ

(0)n+1 − γ

(1)n+1

2n + 1, S(1)

n =n(n + 1)S(0)

n−1 − (n + 1)σ(0)n + γ

(0)n+1 + σ

(1)n+1

2n + 1,

C(0)n =

nσ(0)n − n2S

(0)n−1 + γ

(0)n+1 + σ

(1)n+1

2n + 1, C(1)

n =n(n + 1)C(0)

n−1 − (n + 1)γ(0)n − σ

(0)n+1 + γ

(1)n+1

2n + 1.

101

XXIX. Integrals of the type∫

x2n+1 lnx · Z0(x) dx∫x lnx · J0(x) dx = J0(x) + x lnx · J1(x)∫x lnx · I0(x) dx = −I0(x) + x lnx · I1(x)∫

x3 lnx · J0(x) dx =(x2 − 4 + 2x2 lnx

)J0(x) +

[−4x +

(x3 − 4x

)lnx]

J1(x)∫x3 lnx · I0(x) dx =

(−x2 − 4− 2x2 lnx

)I0(x) +

[4x +

(x3 + 4x

)lnx]

I1(x)∫x5 lnx · J0(x) dx =

=[x4 − 32 x2 + 64 +

(4 x4 − 32 x2

)lnx]

J0(x) +[−8 x3 + 96 x +

(x5 − 16 x3 + 64 x

)lnx]


=[−x4 − 32 x2 − 64 +

(−4 x4 − 32 x2

)lnx]

I0(x) +[8 x3 + 96 x +

(x5 + 16 x3 + 64 x

)lnx]


[x6 − 84 x4 + 1536 x2 − 2304 +

(6 x6 − 144 x4 + 1152 x2

)lnx]

J0(x)+

+[−12 x5 + 480 x3 − 4224 x +

(x7 − 36 x5 + 576 x3 − 2304 x

)lnx]


[−x6 − 84 x4 − 1536 x2 − 2304 +

(−6 x6 − 144 x4 − 1152 x2

)lnx]

I0(x)+

+[12 x5 + 480 x3 + 4224 x +

(x7 + 36 x5 + 576 x3 + 2304 x

)lnx]


=[x8 − 160 x6 + 7680 x4 − 116736 x2 + 147456 +

(8 x8 − 384 x6 + 9216 x4 − 73728 x2

)lnx]

J0(x)+

+[−16 x7 + 1344 x5 − 39936 x3 + 307200 x +

(x9 − 64 x7 + 2304 x5 − 36864 x3 + 147456 x

)lnx]


=[−x8 − 160 x6 − 7680 x4 − 116736 x2 − 147456 +

(−8 x8 − 384 x6 − 9216 x4 − 73728 x2

)lnx]

I0(x)+

+[16 x7 + 1344 x5 + 39936x3 + 307200 x +

(x9 + 64 x7 + 2304 x5 + 36864x3 + 147456 x

)lnx]

I1(x)

Let ∫xn lnx · J0(x) dx = (Pn(x) + Qn(x) ln x)J0(x) + (Rn(x) + Sn(x) ln x)J1(x) ,∫xn lnx · I0(x) dx = (P ∗n(x) + Q∗n(x) ln x)I0(x) + (R∗n(x) + S∗n(x) ln x)I1(x) ,

then holds:P11 = x10 − 260 x8 + 23680x6 − 952320 x4 + 13148160x2 − 14745600

Q11 = 10 x10 − 800 x8 + 38400 x6 − 921600 x4 + 7372800 x2

R11 = −20 x9 + 2880 x7 − 180480 x5 + 4730880x3 − 33669120 x

S11 = x11 − 100 x9 + 6400 x7 − 230400 x5 + 3686400x3 − 14745600 x

P ∗11 = −x10 − 260 x8 − 23680 x6 − 952320 x4 − 13148160 x2 − 14745600

Q∗11 = −10 x10 − 800 x8 − 38400 x6 − 921600 x4 − 7372800 x2

R∗11 = 20 x9 + 2880 x7 + 180480 x5 + 4730880x3 + 33669120 x

S∗11 = x11 + 100 x9 + 6400 x7 + 230400 x5 + 3686400x3 + 14745600x

102

P13 = x12 − 384 x10 + 56640 x8 − 4331520 x6 + 159252480x4 − 2070282240x2 + 2123366400

Q13 = 12 x12 − 1440 x10 + 115200 x8 − 5529600x6 + 132710400x4 − 1061683200 x2

R13 = −24 x11 + 5280 x9 − 568320 x7 + 31518720 x5 − 769720320x3 + 5202247680 x

S13 = x13 − 144 x11 + 14400 x9 − 921600 x7 + 33177600 x5 − 530841600 x3 + 2123366400x

P ∗13 = −x12 − 384 x10 − 56640 x8 − 4331520 x6 − 159252480x4 − 2070282240 x2 − 2123366400

Q∗13 = −12 x12 − 1440 x10 − 115200 x8 − 5529600 x6 − 132710400x4 − 1061683200 x2

R∗13 = 24 x11 + 5280 x9 + 568320 x7 + 31518720 x5 + 769720320x3 + 5202247680x

S∗13 = x13 + 144 x11 + 14400 x9 + 921600 x7 + 33177600 x5 + 530841600x3 + 2123366400x

P15 = x14−532 x12+115584x10−14327040 x8+1003806720x6−34929377280x4+435502448640 x2−416179814400

Q15 = 14x14 − 2352 x12 + 282240 x10 − 22579200 x8 + 1083801600x6 − 26011238400x4 + 208089907200x2

R15 = −28 x13 + 8736 x11 − 1438080 x9 + 137195520x7 − 7106641920 x5 + 165728747520x3 − 1079094804480x

S15 = x15−196 x13+28224 x11−2822400 x9+180633600 x7−6502809600 x5+104044953600 x3−416179814400 x

P ∗15 = −x14−532 x12−115584 x10−14327040 x8−1003806720 x6−34929377280x4−435502448640 x2−416179814400

Q∗15 = −14 x14 − 2352 x12 − 282240 x10 − 22579200 x8 − 1083801600 x6 − 26011238400x4 − 208089907200 x2

R∗15 = 28x13 + 8736 x11 + 1438080 x9 + 137195520x7 + 7106641920 x5 + 165728747520x3 + 1079094804480x

S∗15 = x15+196x13+28224x11+2822400 x9+180633600 x7+6502809600 x5+104044953600 x3+416179814400 x

103

XXX. Integrals of the type∫

x2n lnx · Z1(x) dx∫lnx · J1(x) dx = − lnx · J0(x) +

∫J0(x)

xdx∫

lnx · I1(x) dx = lnx · I0(x)−∫

I0(x)x

dx

Concerning the right integrals see III, page 6.∫x2 lnx · J1(x) dx =

(2− x2 lnx

)J0(x) + x (1 + 2 ln x) J1(x)

∫x2 lnx · I1(x) dx =

(2 + x2 lnx

)I0(x)− x (1 + 2 ln x) I1(x)∫

x4 lnx · J1(x) dx =[6 x2 − 16 +

(−x4 + 8 x2

)lnx]

J0(x) +[x3 − 20 x +

(4 x3 − 16 x

)lnx]


[6 x2 + 16 +

(x4 + 8 x2

)lnx]

I0(x) +[−x3 − 20 x +

(−4 x3 − 16 x

)lnx]


[10 x4 − 224 x2 + 384 +

(−x6 + 24 x4 − 192 x2

)lnx]

J0(x)+

+[x5 − 64 x3 + 640 x +

(6 x5 − 96 x3 + 384 x

)lnx]


[10 x4 + 224 x2 + 384 +

(x6 + 24 x4 + 192 x2

)lnx]

I0(x)+

+[−x5 − 64 x3 − 640 x +

(−6 x5 − 96 x3 − 384 x

)lnx]

I1(x)∫x8 lnx·J1(x) dx =

[14 x6 − 816 x4 + 13440 x2 − 18432 +

(−x8 + 48 x6 − 1152 x4 + 9216 x2

)lnx]

J0(x)+

+[x7 − 132 x5 + 4416 x3 − 36096 x +

(8 x7 − 288 x5 + 4608 x3 − 18432 x

)lnx]


[14 x6 + 816 x4 + 13440 x2 + 18432 +

(x8 + 48 x6 + 1152 x4 + 9216 x2

)lnx]

I0(x)+

+[−x7 − 132 x5 − 4416 x3 − 36096 x +

(−8 x7 − 288 x5 − 4608 x3 − 18432 x

)lnx]

I1(x)

Let ∫xn lnx · J1(x) dx = [Pn(x) + Qn(x) lnx]J0(x) + [Rn(x) + Sn(x) lnx]J1(x) ,∫xn lnx · I1(x) dx = [P ∗n(x) + Q∗n(x) ln x]I0(x) + [R∗n(x) + S∗n(x) ln x]I1(x) ,

then holds:P10(x) = 18 x8 − 1984 x6 + 86016 x4 − 1241088x2 + 1474560

Q10(x) = −x10 + 80 x8 − 3840 x6 + 92160 x4 − 737280 x2

R10(x) = x9 − 224 x7 + 15744 x5 − 436224 x3 + 3219456 x

S10(x) = 10 x9 − 640 x7 + 23040 x5 − 368640 x3 + 1474560 x

P ∗10(x) = 18 x8 + 1984 x6 + 86016 x4 + 1241088x2 + 1474560

Q∗10(x) = x10 + 80 x8 + 3840 x6 + 92160 x4 + 737280 x2

R∗10(x) = −x9 − 224 x7 − 15744 x5 − 436224 x3 − 3219456 x

S∗10(x) = −10 x9 − 640 x7 − 23040 x5 − 368640 x3 − 1474560 x

P12(x) = 22 x10 − 3920 x8 + 322560 x6 − 12349440 x4 + 165150720x2 − 176947200

Q12(x) = −x12 + 120 x10 − 9600 x8 + 460800 x6 − 11059200 x4 + 88473600 x2

104

R12(x) = x11 − 340 x9 + 40960 x7 − 2396160x5 + 60456960x3 − 418775040x

S12(x) = 12 x11 − 1200 x9 + 76800x7 − 2764800 x5 + 44236800 x3 − 176947200x

P ∗12(x) = 22 x10 + 3920 x8 + 322560 x6 + 12349440x4 + 165150720 x2 + 176947200

Q∗12(x) = x12 + 120 x10 + 9600 x8 + 460800 x6 + 11059200x4 + 88473600 x2

R∗12(x) = −x11 − 340 x9 − 40960 x7 − 2396160 x5 − 60456960 x3 − 418775040 x

S∗12(x) = −12 x11 − 1200 x9 − 76800 x7 − 2764800x5 − 44236800 x3 − 176947200x

P14(x) = 26 x12 − 6816 x10 + 908160 x8 − 66170880 x6 + 2362245120x4 − 30045634560x2 + 29727129600

Q14(x) = −x14 + 168 x12 − 20160 x10 + 1612800x8 − 77414400 x6 + 1857945600x4 − 14863564800x2

R14(x) = x13 − 480 x11 + 88320 x9 − 8878080 x7 + 474439680x5 − 11306926080x3 + 74954833920x

S14(x) = 14 x13 − 2016 x11 + 201600 x9 − 12902400 x7 + 464486400x5 − 7431782400 x3 + 29727129600x

P ∗14(x) = 26 x12 + 6816 x10 + 908160 x8 + 66170880x6 + 2362245120x4 + 30045634560x2 + 29727129600

Q∗14(x) = x14 + 168 x12 + 20160 x10 + 1612800 x8 + 77414400 x6 + 1857945600x4 + 14863564800x2

R∗14(x) = −x13 − 480 x11 − 88320 x9 − 8878080 x7 − 474439680x5 − 11306926080x3 − 74954833920x

S∗14(x) = −14 x13 − 2016 x11 − 201600 x9 − 12902400 x7 − 464486400 x5 − 7431782400x3 − 29727129600x

105

XXXI. Integrals of the type∫

x2n+ν lnx · Zν(x) dx

a) The Functions Λk and Λ∗k, k = 0, 1 :

Let

Λ0(x) =∞∑

k=0

αk x2k+1 =∞∑

k=0

(−1)k

22k · (k!)2 · (2k + 1)x2k+1 =

∫ x

0

J0(t) dt = xJ0(x) + Φ(x) .

(Φ(x) and further on Ψ(x) defined as on page 3)

Λ0(x) = x− x3

12+

x5

320− x7

16 128+

x9

1 327 104− x11

162 201 600+

x13

27 603 763 200− x15

6 242 697 216 000+ . . .

k αk 1/αk

0 1.0000000000000000E+00 11 -8.3333333333333329E-02 -122 3.1250000000000002E-03 3203 -6.2003968253968251E-05 -161284 7.5352044753086416E-07 13271045 -6.1651672979797980E-09 -1622016006 3.6226944592830012E-11 276037632007 -1.6018716996829596E-13 -62426972160008 5.5211570531352012E-16 18112145522688009 -1.5246859958300586E-18 -655872751986278400

10 3.4486945143775135E-21 28996479561498624000011 -6.5058017249306315E-24 -15370895737076318208000012 1.0391211088430869E-26 9623517331039086182400000013 -1.4232975959389203E-29 -7025937533045016040046592000014 1.6902284962328838E-32 5916359842641166099499974656000015 -1.7568683294176930E-35 -5691946193437535661243079065600000016 1.6117104111016952E-38 -5691946193437535661243079065600000017 -1.3145438350557573E-41 -7607201626392202499431885757743104000000018 9.5948102742224525E-45 10422300925393111264364508167294209228800000019 -6.3038564554696844E-48 -15863305376065904161187882214847045592678400000020 3.7477195390749646E-51 266828931453826490506134634177940048943513600000000

Values of this function may be found in [1], Table 11.1 .Asymptotic expansion:

Λ0(x) ∼ 1 +

√2

πx

∞∑k=0

λk

xksin(

x +2k − 1

4π

)Recurrence relation:

λk+1 = − 2k + 116(k + 1)

[(12k + 10)λk + (4k2 − 1)λk−1

]If k > 1, then up to k ≈ 30 holds

λk ≈ (−1)k Γ(sk) with sk = k +12− 1

3√

k.

106

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0.2

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0.6

0.8

1.0

1.2

1.4

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Λ0(x)

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Figure 16 : Function Λ0 (x)

Coefficients of the asymptotic formula:

k λk λk qk = |λk/λk−1| |λk|/Γ(sk)

0 1 1 - -1 −5

8-0.625 0.625 -

2 129128

1.007812500 1.612500000 0.882203509

3 −26551024

-2.592773438 2.572674419 0.958684418

4 30103532768

9.186859131 3.543255650 0.992044824

5 −10896795262144

-41.56797409 4.524720963 1.007474317

6 9613192054194304

229.1963589 5.513772656 1.014554883

7 −5004657157533554432

-1491.504060 6.507538198 1.017456390

8 240353982618752147483648

11192.35450 7.504072427 1.018151550

9 −163482593611837517179869184

-95159.39374 8.502178320 1.017633944

10 248523783571238175274877906944

904124.2577 9.501156135 1.016430975

11 −208772102204411996252199023255552

-9493856.042 10.50060980 1.014835786

12 768302714773631314777570368744177664

109182382.6 11.50032001 1.014835786

107

Roughly spoken, the itemλk

xksin(

x +2k − 1

4π

)in the asymptotic series should not be used if |x| < qk.

Let

dn(x) = 1 +

√2

πx

n∑k=0

λk

xksin(

x +2k − 1

4π

)− Λ0(x) .

The following table gives some consecutive maxima and minima of interest of this functions:

n = 0 n = 1 n = 2 n = 3x dn (x) x dn (x) x dn (x) x dn (x)

3.953 -5.390E-2 5.510 -9.219-3 3.936 1.020E-2 5.501 2.128E-37.084 2.479E-2 8.647 3.315E-3 7.074 -1.751-3 8.642 -3.501E-410.221 -1.475E-2 11.787 -1.592E-3 10.214 5.360E-4 11.783 9.561E-513.360 1.000E-2 14.927 9.002E-4 13.355 -2.197E-4 14.924 -3.469E-516.500 -7.337E-3 18.068 -5.647E-4 16.496 1.076E-4 18.065 1.511E-5

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 2 3 4 7

d0(x)

d1(x)d2(x)

d3(x)

Figure 16 : Differences d0 (x) . . . d3 (x)

Let

Λ∗0(x) =∞∑

k=0

|αk|x2k+1 =∞∑

k=0

122k · (k!)2 · (2k + 1)

x2k+1 =∫ x

0

I0(t) dt = x I0(x) + Ψ(x) .

Asymptotic expansion (see Λ0 (x)):

Λ∗0 (x) =ex

√2πx

[1 +

58x

+129

128x2+

26551024x3

+ . . .

]

108

Furthermore, let

Λ1(x) =∞∑

k=0

βk x2k+1 =∞∑

k=0

(−1)k

22k · (k!)2 · (2k + 1)2x2k+1 =

∞∑k=0

αk

2k + 1x2k+1 .

Λ1(x) can be written as a hypergeometric function.One has

Λ0(x) = xΛ′1(x) , Λ1(0) = 0 ⇐⇒ Λ1 (x) =∫ x

0

Λ0 (t) dt

t.

Λ1(x) = x− x3

36+

x5

1 600− x7

112 896+

x9

11 943 936− x11

1 784 217 600+

x13

358 848 921 600− x15

93 640 458 240 000+ . . .

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0.6

1.2

1.8

2.4

3.0

3.6

4.2

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Λ1(x)

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. . . . . . .. . . . . . . .

Figure 17 : Function Λ1 (x), dots: C + ln 2x

Asymptotic series with Euler’s constant C = 0.577 215 664 901 533 :

Λ1 (x) ∼ C + ln 2x−√

2πx

∞∑k=1

µk

xksin(

x +2k − 1

4π

)

109

k µk µk |µk/µk−1|1 1 1 1

2 −178

-2.125 2.125

3 809128

6.320 312 500 2.974

4 −253071024

-24.713 867 187 500 3.910

5 394524332768

120.399 261 474 609 4.871

6 −184487487262144

-703.763 912 200 928 5.845

7 201480178534194304

4 803.661 788 225 174 6.826

8 −125892764275533554432

-37 518.967 472 166 7.810

9 7088920359205952147483648

330 103.578 008 988 8.798

10 −5551062066608359517179869184

-3 231 143.384 827 510 9.788

11 9574308055473282135274877906944

34 831 129.798 379 10.780

12 −9017135513239831560452199023255552

-410 051 848.723 007 11.773

Some consecutive maxima and minima of the differences

δn(x) = C + ln 2x−√

2πx

n∑k=1

µk

xksin(

x +3− 2k

4π

)− Λ1 (x)

n = 1 x 2.4704 5.569 8.689 11.819 14.953 18.090

δn(x) 9.374E-2 -1.855E-2 6.824E-3 -3.309E-3 1.880E-3 -1.182E-3

n = 2 x 3.991 7.115 10.246 13.380 16.517 19.655

δn(x) 2.312E-2 -4.117E-3 1.279E-3 -5.284E-4 2.598E-4 -1.436E-4

n = 3 x 5.541 8.673 11.808 14.945 18.083 21.222

δn(x) 5.439E-3 -9.148E-4 2.525E-4 -9.214E-5 4.028E-5 -1.997E-5

n = 4 x 10.237 13.374 16.512 19.651 22.791 25.931

δn(x) -2.043E-4 5.163E-5 -1.707E-5 6.771E-6 -3.061E-6 1.527E-6

Let

Λ∗1(x) =∞∑

k=0

|βk|x2k+1 =∞∑

k=0

x2k+1

22k · (k!)2 · (2k + 1)2.

b) Basic Integrals: ∫lnx · J0(x) dx = Λ0 (x) · lnx− Λ1 (x) + c∫lnx · I0(x) dx = Λ∗0 (x) · lnx− Λ∗1 (x) + c

In particular, let ∫ x

0

ln t · J0(t) dt = F (x) and∫ x

0

ln t · I0(t) dt = F ∗ (x) .

110

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0.2

0.4

0.6

-0.2

-0.4

-0.6

-0.8

-1.0

-1.2

-1.4

-1.6

-1.8

.............2

.............4

.............6

.............8

.............10

.............12

.............14

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F (x)

F ∗(x)

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

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Figure 18 : Functions F (x) and F ∗ (x)

Holds ([7 ], 6.772) with Euler’s constant C = 0.577...

limx→∞

F (x) = − ln 2−C = −1.270 362 845 461 478 170 .

Asymptotic expansion:

F (x) ∼ − ln 2−C +

√2

πx

[lnx · sin

(x− π

4

)+

∞∑k=1

λk lnx + µk

xk· sin

(x +

(2k − 1)π4

)]=

= − ln 2−C +

√2

πx

[lnx · sin

(x− π

4

)− 5 ln x + 8

8xsin(x +

π

4

)+

129 ln x + 272128x2

sin(

x +3π

4

)−

−2 655 lnx + 6 4721 024x3

sin(

x +5π

4

)+

301 035 lnx + 809 82432 768x4

sin(

x +7π

4

)+ . . .

]Let

∆n(x) = − ln 2−C +

√2

πx

[lnx · sin

(x− π

4

)+

n∑k=1

λk lnx + µk

xk· sin

(x +

(2k − 1)π4

)]− F (x)

111

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-5

-10

5

10

15

20

25

............7.5

............10 x

............12.5

∆n(x)

............15

............17.5

............................................................................................................................................................................................................................ 1000 ∆1(x)

.................................... .................................... .................................... ....... 1000 ∆1(x)

................. ................. ................. ................. ................. ................. ................. 1000 ∆3(x)

. . . . . . . . . . . . . . . . . . 1000 ∆4(x)

Figure 19 : ∆n(x), n = 1− 4, 5 ≤ x ≤ 20

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. . . . . . . . . 105 ∆4(x)

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Figure 20 : ∆n(x), n = 2− 4, 10 ≤ x ≤ 30

112

Some consecutive maxima and minima of the differences ∆n(x):

n = 1 x 11.822 14.944 18.079 21.217 24.356 27.496

∆n(x) -6.213E-4 5.541E-4 -4.520E-4 3.644E-4 -2.959E-4 2.431E-4

n = 2 x 7.046 10.134 13.404 16.513 19.647 22.785

∆n(x) 6.878E-4 -3.344E-5 -4.133E-5 4.183E-5 -3.303E-5 2.498E-5

n = 3 x 18.083 21.215 24.354 27.494 30.634 33.775

∆n(x) 3.458-6 -2.863E-6 2.100E-6 -1.507E-6 1.087E-6 -7.949E-7

n = 4 x 22.785 25.923 29.063 32.204 35.345 38.486

∆n(x) -2.648E-7 1.982E-7 -1.385E-7 9.570E-8 -6.666E-8 4.709E-8

c) Integrals of x2n lnx · Z0(x):

∫lnx · J0(x) dx = [xJ0(x) + Φ(x)] · lnx− Λ1(x)∫lnx · I0(x) dx = [x I0(x) + Ψ(x)] · lnx− Λ∗1(x)∫

x2 lnx · J0(x) dx = [x2 J1(x)− Φ(x)] · lnx− xJ0(x) + Λ1(x)− 2Φ(x)∫x2 lnx · I0(x) dx = [x2 I1(x) + Ψ(x)] · lnx + xI0(x)− Λ∗1(x) + 2Ψ(x)∫

x4 lnx · J0(x) dx =

= [3 x3 J0(x) + (x4 − 9 x2) J1(x) + 9Φ(x)] · lnx + (x3 + 9 x) J0(x)− 6 x2 J1(x)− 9Λ1(x) + 24Φ(x)∫x4 lnx · I0(x) dx =

= [−3 x3 I0(x) + (x4 + 9 x2) I1(x) + 9Ψ(x)] · lnx + (−x3 + 9 x) I0(x) + 6 x2 I1(x)− 9Λ∗1(x) + 24Ψ(x)

Let ∫xn lnx · J0(x) dx =

= [Pn(x) J0(x) + Qn(x) J1(x) + pn Φ(x)] · lnx + Rn(x) J0(x) + Sn(x) J1(x)− pn Λ1(x) + qn Φ(x)

and ∫xn lnx · I0(x) dx =

= [P ∗n(x) I0(x) + Q∗n(x) I1(x) + p∗n Ψ(x)] · lnx + R∗n(x) I0(x) + S∗n(x) I1(x)− p∗n Λ∗1(x) + q∗n Ψ(x) ,

then holds

P6(x) = 5 x5 − 75 x3 , Q6(x) = x6 − 25 x4 + 225 x2 , R6(x) = x5 − 55 x3 − 225 x

S6(x) = −10 x4 + 240 x2 , p6 = −225 , q6 = −690

P ∗6 (x) = −5 x5 − 75 x3 , Q∗6(x) = x6 + 25 x4 + 225 x2 , R∗6(x) = −x5 − 55 x3 + 225 x

S∗6 (x) = 10 x4 + 240 x2 , p∗6 = 225 , q∗6 = 690

P8(x) = 7 x7 − 245 x5 + 3675 x3 , Q8(x) = x8 − 49 x6 + 1225 x4 − 11025 x2

R8(x) = x7−119 x5+3745x3+11025x , S8(x) = −14 x6+840x4−14910 x2 , p8 = 11025 , q8 = 36960

P ∗8 (x) = −7 x7 − 245 x5 − 3675 x3 , Q∗8(x) = x8 + 49 x6 + 1225 x4 + 11025 x2

R∗8(x) = −x7−119 x5−3745 x3+11025 x , S∗8 (x) = 14 x6+840 x4+14910 x2 , p∗8 = 11025 , q∗8 = 36960

113

P10(x) = 9 x9 − 567 x7 + 19845 x5 − 297675 x3 , Q10(x) = x10 − 81 x8 + 3969 x6 − 99225 x4 + 893025 x2

R10(x) = x9−207 x7+14049 x5−369495 x3−893025 x , S10(x) = −18 x8+2016 x6−90090 x4+1406160 x2

p10 = −893025 , q10 = −3192210

P ∗10(x) = −9 x9 − 567 x7 − 19845 x5 − 297675 x3 , Q∗10(x) = x10 + 81x8 + 3969x6 + 99225x4 + 893025x2

R∗10(x) = −x9−207 x7−14049 x5−369495 x3+893025 x , S∗10(x) = 18 x8+2016 x6+90090 x4+1406160 x2

p∗10 = 893025 , q∗10 = 3192210

P12(x) = 11 x11 − 1089 x9 + 68607x7 − 2401245 x5 + 36018675 x3

Q12(x) = x12 − 121 x10 + 9801 x8 − 480249 x6 + 12006225 x4 − 108056025x2

R12(x) = x11 − 319 x9 + 37521 x7 − 2136519 x5 + 51257745 x3 + 108056025x

S12(x) = −22 x10 + 3960 x8 − 331254 x6 + 13083840 x4 − 189791910 x2

p12 = 108056025 , q12 = 405903960

P ∗12(x) = −11 x11 − 1089 x9 − 68607 x7 − 2401245 x5 − 36018675 x3

Q∗12(x) = x12 + 121 x10 + 9801 x8 + 480249 x6 + 12006225 x4 + 108056025x2

R∗12(x) = −x11 − 319 x9 − 37521 x7 − 2136519 x5 − 51257745 x3 + 108056025x

Q∗12(x) = 22 x10 + 3960 x8 + 331254 x6 + 13083840 x4 + 189791910x2

p∗12 = 108056025 , q∗12 = 405903960

P14(x) = 13 x13 − 1859 x11 + 184041 x9 − 11594583 x7 + 405810405x5 − 6087156075 x3

Q14(x) = x14 − 169 x12 + 20449x10 − 1656369 x8 + 81162081 x6 − 2029052025x4 + 18261468225x2

R14(x) = x13 − 455 x11 + 82225x9 − 8124831 x7 + 423504081x5 − 9599044455x3 − 18261468225x

S14(x) = −26 x12 + 6864 x10 − 924066 x8 + 68468400x6 − 2523330810 x4 + 34884289440x2

p14 = −18261468225 , q14 = −71407225890

P ∗14(x) = −13 x13 − 1859 x11 − 184041 x9 − 11594583 x7 − 405810405x5 − 6087156075 x3

Q∗14(x) = x14 + 169 x12 + 20449x10 + 1656369x8 + 81162081 x6 + 2029052025x4 + 18261468225x2

R∗14(x) = −x13 − 455 x11 − 82225 x9 − 8124831 x7 − 423504081x5 − 9599044455 x3 + 18261468225x

S∗14(x) = 26 x12 + 6864 x10 + 924066 x8 + 68468400 x6 + 2523330810x4 + 34884289440x2

p∗14 = 18261468225 , q∗14 = 71407225890

P16(x) = 15 x15−2925 x13+418275 x11−41409225 x9+2608781175 x7−91307341125x5+1369610116875 x3

Q16(x) = x16 − 225 x14 + 38025x12 − 4601025 x10 + 372683025x8 − 18261468225x6+

+456536705625x4 − 4108830350625x2

R16(x) = x15 − 615 x13 + 158145 x11 − 24021855 x9 + 2175924465x7 − 107462730375 x5+

+2342399684625x3 + 4108830350625x

S16(x) = −30 x14 + 10920 x12 − 2157870 x10 + 257605920x8 − 17840252430x6+

+628620993000x4 − 8396809170750x2

p16 = 4108830350625 , q16 = 16614469872000

114

P ∗16(x) = −15 x15−2925 x13−418275 x11−41409225 x9−2608781175 x7−91307341125x5−1369610116875x3

Q∗16(x) = x16 + 225 x14 + 38025 x12 + 4601025 x10 + 372683025x8 + 18261468225x6+

+456536705625x4 + 4108830350625x2

R∗16(x) = −x15 − 615 x13 − 158145 x11 − 24021855 x9 − 2175924465 x7 − 107462730375 x5−

−2342399684625x3 + 4108830350625x

S∗16(x) = 30 x14 + 10920x12 + 2157870x10 + 257605920x8 + 17840252430x6+

+628620993000x4 + 8396809170750x2

p∗16 = 4108830350625 , q∗16 = 16614469872000

P18(x) = 17 x17 − 4335 x15 + 845325 x13 − 120881475x11 + 11967266025x9 − 753937759575x7+

+26387821585125x5 − 395817323776875x3

Q18(x) = x18 − 289 x16 + 65025x14 − 10989225 x12 + 1329696225x10 − 107705394225 x8+

+5277564317025x6 − 131939107925625x4 + 1187451971330625x2

R18(x) = x17 − 799 x15 + 277185 x13 − 59925255 x11 + 8350229745 x9 − 717540730335x7+

+34161178676625x5 − 723520252830375x3 − 1187451971330625x

S18(x) = −34 x16 + 16320x14 − 4448730 x12 + 780059280x10 − 87119333730x8 + 5776722871920x6−

−197193714968250x4 + 2566378082268000x2

p18 = −1187451971330625 , q18 = −4941282024929250

P ∗18(x) = −17 x17 − 4335 x15 − 845325 x13 − 120881475 x11 − 11967266025x9 − 753937759575 x7−

−26387821585125x5 − 395817323776875x3

Q∗18(x) = x18 + 289 x16 + 65025 x14 + 10989225 x12 + 1329696225x10 + 107705394225x8+

+5277564317025x6 + 131939107925625x4 + 1187451971330625x2

R∗18(x) = −x17 − 799 x15 − 277185 x13 − 59925255 x11 − 8350229745x9 − 717540730335x7−

−34161178676625x5 − 723520252830375x3 + 1187451971330625x

S∗18(x) = 34 x16 + 16320 x14 + 4448730x12 + 780059280x10 + 87119333730x8 + 5776722871920x6+

+197193714968250x4 + 2566378082268000x2

p∗18 = 1187451971330625 , q∗18 = 4941282024929250

d) Integrals of x2n+1 lnx · Z1(x):

∫x lnx · J1(x) dx = Φ(x) · lnx + xJ0(x)− Λ1(x) + Φ(x)∫

x lnx · I1(x) dx = −Ψ(x) · lnx− x I0(x) + Λ∗1(x)−Ψ(x)∫x3 lnx · J1(x) dx = [−x3 J0(x) + 3x2 J1(x)− 3 Φ(x)] lnx− 3xJ0(x) + x2 J1(x) + 3 Λ1(x)− 7 Φ(x)∫x3 lnx · I1(x) dx = [x3 I0(x)− 3x2 I1(x)− 3 Ψ(x)] lnx− 3x I0(x)− x2 I1(x) + 3 Λ∗1(x)− 7 Ψ(x)

Let ∫xn lnx · J0(x) dx =

115

= [Pn(x) J0(x) + Qn(x) J1(x) + pn Φ(x)] · lnx + Rn(x) J0(x) + Sn(x) J1(x)− pn Λ1(x) + qn Φ(x)

and ∫xn lnx · I0(x) dx =

= [P ∗n(x) I0(x) + Q∗n(x) I1(x) + p∗n Ψ(x)] · lnx + R∗n(x) I0(x) + S∗n(x) I1(x)− p∗n Λ∗1(x) + q∗n Ψ(x) ,

then holds

P5(x) = −x5 + 15 x3 , Q5(x) = 5 x4 − 45 x2 , R5(x) = 8 x3 + 45 x , S5(x) = x4 − 39 x2 ,

p5 = 45 , q5 = 129

P ∗5 (x) = x5 + 15 x3 , Q∗5(x) = −5 x4 − 45 x2 , R∗5(x) = 8 x3 − 45 x , S∗5 (x) = −x4 − 39 x2

p∗5 = −45 , q∗5 = −129

P7(x) = −x7 + 35 x5 − 525 x3 , Q7(x) = 7 x6 − 175 x4 + 1575 x2

R7(x) = 12 x5 − 460 x3 − 1575 x , S7(x) = x6 − 95 x4 + 1905 x2 , p7 = −1575 , q7 = −5055

P ∗7 (x) = x7 + 35 x5 + 525 x3 , Q∗7(x) = −7 x6 − 175 x4 − 1575 x2

R∗7(x) = 12 x5 + 460 x3 − 1575 x , S∗7 (x) = −x6 − 95 x4 − 1905 x2 , p∗7 = −1575 , q∗7 − 5055

P9(x) = −x9 + 63 x7 − 2205 x5 + 33075 x3 , Q9(x) = 9 x8 − 441 x6 + 11025 x4 − 99225 x2

R9(x) = 16 x7 − 1316 x5 + 37380 x3 + 99225 x , S9(x) = x8 − 175 x6 + 8785 x4 − 145215 x2

p9 = 99225 , q9 = 343665

P ∗9 (x) = x9 + 63 x7 + 2205 x5 + 33075 x3 , Q∗9(x) = −9 x8 − 441 x6 − 11025 x4 − 99225 x2

R∗9(x) = 16 x7 + 1316 x5 + 37380 x3 − 99225 x , S∗9 (x) = −x8 − 175 x6 − 8785 x4 − 145215 x2

p∗9 = −99225 , q∗9 = −343665

P11(x) = −x11 + 99 x9 − 6237 x7 + 218295 x5 − 3274425 x3

Q11(x) = 11 x10 − 891 x8 + 43659 x6 − 1091475 x4 + 9823275 x2

R11(x) = 20 x9 − 2844 x7 + 174384 x5 − 4362120 x3 − 9823275x

S11(x) = x10 − 279 x8 + 26145 x6 − 1090215 x4 + 16360785x2

p11 = −9823275 , q11 = −36007335

P ∗11(x) = x11 + 99 x9 + 6237 x7 + 218295 x5 + 3274425x3

Q∗11(x) = −11 x10 − 891 x8 − 43659 x6 − 1091475 x4 − 9823275 x2

R∗11(x) = 20 x9 + 2844 x7 + 174384 x5 + 4362120x3 − 9823275x

R∗11(x) = −x10 − 279 x8 − 26145 x6 − 1090215 x4 − 16360785 x2

p∗11 = −9823275 , q∗11 = −36007335

P13(x) = −x13 + 143 x11 − 14157 x9 + 891891 x7 − 31216185 x5 + 468242775x3

Q13(x) = 13 x12 − 1573 x10 + 127413 x8 − 6243237 x6 + 156080925x4 − 1404728325 x2

R13(x) = 24 x11 − 5236 x9 + 556380 x7 − 30175992 x5 + 702369360x3 + 1404728325 x

S13(x) = x12 − 407 x10 + 61281 x8 − 4786551 x6 + 182096145x4 − 2575350855x2

116

p13 = 1404728325 , q13 = 5384807505

P ∗13(x) = x13 + 143 x11 + 14157x9 + 891891 x7 + 31216185x5 + 468242775 x3

R∗13(x) = −13 x12 − 1573 x10 − 127413 x8 − 6243237x6 − 156080925x4 − 1404728325 x2

R∗13(x) = 24 x11 + 5236 x9 + 556380 x7 + 30175992 x5 + 702369360x3 − 1404728325x

S∗13(x) = −x12 − 407 x10 − 61281 x8 − 4786551 x6 − 182096145x4 − 2575350855x2

p∗13 = −1404728325 , q∗13 = −5384807505

P15(x) = −x15 + 195 x13 − 27885 x11 + 2760615x9 − 173918745 x7+

+6087156075x5 − 91307341125 x3

Q15(x) = 15 x14 − 2535 x12 + 306735 x10 − 24845535 x8 + 1217431215x6−

−30435780375x4 + 273922023375x2

R15(x) = 28 x13 − 8684 x11 + 1417416 x9 − 133467048 x7 + 6758371620x5−

−150072822900x3 − 273922023375x

S15(x) = x14 − 559 x12 + 123409 x10 − 15517359 x8 + 1108188081 x6−

−39879014175x4 + 541525809825x2

p15 = −273922023375 , q15 = −1089369856575

P ∗15(x) = x15 + 195 x13 + 27885x11 + 2760615x9 + 173918745x7+

+6087156075x5 + 91307341125x3

Q∗15(x) = −15 x14 − 2535 x12 − 306735 x10 − 24845535 x8 − 1217431215 x6−

−30435780375x4 − 273922023375x2

R∗15(x) = 28 x13 + 8684 x11 + 1417416 x9 + 133467048x7 + 6758371620x5+

+150072822900x3 − 273922023375x

S∗15(x) = −x14 − 559 x12 − 123409 x10 − 15517359 x8 − 1108188081 x6−

−39879014175x4 − 541525809825x2

p∗15 = −273922023375 , q∗15 = −1089369856575

P17(x) = −x17 + 255 x15 − 49725 x13 + 7110675x11 − 703956825 x9 + 44349279975x7−

−1552224799125x5 + 23283371986875x3

Q17(x) = 17 x16 − 3825 x14 + 646425 x12 − 78217425 x10 + 6335611425x8 − 310444959825x6+

+7761123995625x4 − 69850115960625x2

R17(x) = 32 x15 − 13380 x13 + 3106740x11 − 449780760 x9 + 39599497080x7 − 1918173757500x5+

+41190404755500x3 + 69850115960625x

S17(x) = x16 − 735 x14 + 223665 x12 − 41284815 x10 + 4751983665x8 − 321545759535 x6+

+11143093586625x4 − 146854586253375x2

p17 = 69850115960625 , q17 = 286554818174625

P ∗17(x) = x17 + 255 x15 + 49725 x13 + 7110675x11 + 703956825x9 + 44349279975x7 + 1552224799125x5+

+23283371986875x3

Q∗17(x) = −17 x16 − 3825 x14 − 646425 x12 − 78217425 x10 − 6335611425x8 − 310444959825 x6−

−7761123995625x4 − 69850115960625x2

117

R∗17(x) = 32 x15 + 13380x13 + 3106740x11 + 449780760x9 + 39599497080x7 + 1918173757500x5+

+41190404755500x3 − 69850115960625x

S∗17(x) = −x16 − 735 x14 − 223665 x12 − 41284815 x10 − 4751983665x8 − 321545759535x6−

−11143093586625 x4 − 146854586253375x2

p∗17 = −69850115960625 , q∗17 = −286554818174625

P19(x) = −x19 + 323 x17 − 82365 x15 + 16061175 x13 − 2296748025 x11 + 227378054475x9−

−14324817431925x7 + 501368610117375x5 − 7520529151760625x3

Q19(x) = 19 x18 − 5491 x16 + 1235475x14 − 208795275x12 + 25264228275x10 − 2046402490275x8+

+100273722023475x6 − 2506843050586875x4 + 22561587455281875x2

R19(x) = 36 x17 − 19516 x15 + 6111840x13 − 1259461320 x11 + 170621631180x9 − 14387211635940 x7+

+675450216441000x5 − 14142702127554000x3 − 22561587455281875x

S19(x) = x18 − 935 x16 + 375105 x14 − 95515095 x12 + 16150822545x10 − 1762972735095x8+

+115035298883505x6 − 3878619692322375x4 + 49948635534422625x2

p19 = −22561587455281875 , q19 = −95071810444986375

P ∗19(x) = x19 + 323 x17 + 82365x15 + 16061175 x13 + 2296748025x11 + 227378054475x9+

+14324817431925x7 + 501368610117375x5 + 7520529151760625x3

Q∗19(x) = −19 x18 − 5491 x16 − 1235475x14 − 208795275 x12 − 25264228275x10 − 2046402490275x8−

−100273722023475x6 − 2506843050586875x4 − 22561587455281875x2

R∗19(x) = 36 x17 + 19516 x15 + 6111840 x13 + 1259461320x11 + 170621631180x9 + 14387211635940x7+

+675450216441000x5 + 14142702127554000x3 − 22561587455281875x

S∗19(x) = −x18 − 935 x16 − 375105 x14 − 95515095 x12 − 16150822545x10 − 1762972735095x8−

−115035298883505x6 − 3878619692322375x4 − 49948635534422625x2

p∗19 = −22561587455281875 , q∗19 = −95071810444986375

118

XXXII. Integrals with Orthogonal Polynomials Fn(x):∫

Fn(x) · Zν(x) dx

The used form of the orthogonal polynomials is shown in the integrals with J0(x).Φ(x) and Ψ(x) are defined on page 3.

a) Legendre Polynomials Pn(x) :

Weight function: w(x) ≡ 1, interval −1 ≤ x ≤ 1.

Holds∫

P0(x) Zν(x) dx =∫

Zν(x) dx and∫

P1(x) Zν(x) dx =∫

xZν(x) dx .The denominator of the polynomial is written in front of the integral on the left hand side.

n = 2

2∫

P2(x) · J0(x) dx =∫

(3x2 − 1) J0(x) dx = −xJ0(x) + 3x2 J1(x)− 4 Φ(x)

2∫

P2(x) · J1(x) dx =(−3x2 + 1

)J0(x) + 6xJ1(x)

2∫

P2(x) · I0(x) dx = −x I0(x) + 3x2 I1(x) + 2 Ψ(x)

2∫

P2(x) · I1(x) dx = (3x2 − 1) I0(x)− 6x I1(x)

n = 3

2∫

P3(x) · J0(x) dx =∫

(5x3 − 3x) J0(x) dx = 10x2J0(x) +(5x3 − 23x

)J1(x)

2∫

P3(x) · J1(x) dx = −5x3 J0(x) + 15x2 J1(x)− 18Φ(x)

2∫

P3(x) · I0(x) dx = −10x2 I0(x) + (5x3 + 17x) I1(x)

2∫

P3(x) · I1(x) dx = 5x3 I0(x)− 15x2 I1(x)− 12 Ψ(x)

n = 4

8∫

P4(x) · J0(x) dx =∫ (

35 x4 − 30 x2 + 3)

J0(x) dx =

=(105 x3 + 3 x

)J0(x) +

(35 x4 − 345 x2

)J1(x) + 348Φ(x)

8∫

P4(x) · J1(x) dx =(−35 x4 + 310 x2 − 3

)J0(x) +

(140 x3 − 620 x

)J1(x)

8∫

P4(x) · I0(x) dx =(−105 x3 + 3 x

)I0(x) +

(35 x4 + 285 x2

)I1(x) + 288 Ψ(x)

8∫

P4(x) · I1(x) dx =(35 x4 + 250 x2 + 3

)I0(x) +

(−140 x3 − 500 x

)I1(x)

n = 5

8∫

P5(x) · J0(x) dx =∫ (

63 x5 − 70 x3 + 15 x)

J0(x) dx =

=(252 x4 − 2156 x2

)J0(x) +

(63 x5 − 1078 x3 + 4327 x

)J1(x)

8∫

P5(x) · J1(x) dx =(−63 x5 + 1015 x3

)J0(x) +

(315 x4 − 3045 x2

)J1(x) + 3060 Φ(x)

8∫

P5(x) · I0(x) dx =(−252 x4 − 1876 x2

)I0(x) +

(63 x5 + 938 x3 + 3767 x

)I1(x)

8∫

P5(x) · I1(x) dx =(63 x5 + 875 x3

)I0(x) +

(−315 x4 − 2625 x2

)I1(x)− 2640 Ψ(x)

n = 6

16∫

P6(x) · J0(x) dx =∫ (

231 x6 − 315 x4 + 105 x2 − 5)

J0(x) dx =

119

=(1155 x5 − 18270 x3 − 5 x

)J0(x) +

(231 x6 − 6090 x4 + 54915 x2

)J1(x)− 54920 Φ(x)

16∫

P6(x) · J1(x) dx =(−231 x6 + 5859 x4 − 46977 x2 + 5

)J0(x) +

(1386 x5 − 23436 x3 + 93954 x

)J1(x)

16∫

P6(x)·I0(x) dx =(−1155 x5 − 16380 x3 − 5 x

)I0(x)+

(231 x6 + 5460 x4 + 49245 x2

)I1(x)+49240Ψ(x)

16∫

P6(x) · I1(x) dx =(231 x6 + 5229 x4 + 41937x2 − 5

)I0(x) +

(−1386 x5 − 20916 x3 − 83874 x

)I1(x)

n = 7

16∫

P7(x) · J0(x) dx =∫ (

429 x7 − 693 x5 + 315 x3 − 35 x)

J0(x) dx =

+(2574 x6 − 64548 x4 + 517014 x2

)J0(x) +

(429 x7 − 16137 x5 + 258507 x3 − 1034063 x

)J1(x)

16∫

P7(x) · J1(x) dx =

=(−429 x7 + 15708 x5 − 235935 x3

)J0(x) +

(3003 x6 − 78540 x4 + 707805 x2

)J1(x)− 707840 Φ(x)

16∫

P7(x) · I0(x) dx =

=(−2574 x6 − 59004 x4 − 472662 x2

)I0(x) +

(429 x7 + 14751 x5 + 236331 x3 + 945289 x

)I1(x)

16∫

P7(x) · I1(x) dx =

=(429 x7 + 14322 x5 + 215145 x3

)I0(x) +

(−3003 x6 − 71610 x4 − 645435 x2

)I1(x)− 645400 Ψ(x)

n = 8

128∫

P8(x) · J0(x) dx =∫ (

6435 x8 − 12012 x6 + 6930 x4 − 1260 x2 + 35)

J0(x) dx =

=(45045 x7 − 1636635 x5 + 24570315 x3 + 35 x

)J0(x)+

+(6435 x8 − 327327 x6 + 8190105x4 − 73712205 x2

)J1(x) + 73712240 Φ(x)

128∫

P8(x) · J1(x) dx =(−6435 x8 + 320892 x6 − 7708338 x4 + 61667964 x2 − 35

)J0(x)+

+(51480 x7 − 1925352 x5 + 30833352 x3 − 123335928x

)J1(x)

128∫

P8(x) · I0(x) dx =(−45045 x7 − 1516515x5 − 22768515 x3 + 35 x

)I0(x)+

+(6435x8 + 303303 x6 + 7589505x4 + 68304285 x2) I1(x) + 68304320 Ψ(x)

128∫

P8(x) · I1(x) dx =(6435 x8 + 296868 x6 + 7131762 x4 + 57052836x2 + 35

)I0(x)+

+(−51480 x7 − 1781208 x5 − 28527048 x3 − 114105672x

)I1(x)

n = 9

128∫

P9(x) · J0(x) dx =∫ (

12155 x9 − 25740 x7 + 18018 x5 − 4620 x3 + 315 x)

J0(x) dx =

=(97240 x8 − 4821960 x6 + 115799112x4 − 926402136x2

)J0(x)+

+(12155 x9 − 803660 x7 + 28949778x5 − 463201068 x3 + 1852804587x

)J1(x)

128∫

P9(x) · J1(x) dx =(−12155 x9 + 791505 x7 − 27720693 x5 + 415815015x3

)J0(x)+

+(109395 x8 − 5540535 x6 + 138603465x4 − 1247445045x2

)J1(x) + 1247445360 Φ(x)

128∫

P9(x) · I0(x) dx =(−97240 x8 − 4513080x6 − 108385992 x4 − 867078696 x2

)I0(x)+

120

+(12155 x9 + 752180 x7 + 27096498 x5 + 433539348x3 + 1734157707 x

)I1(x)

128∫

P9(x) · I1(x) dx =(12155 x9 + 740025 x7 + 25918893 x5 + 388778775x3

)I0(x)+

+(−109395 x8 − 5180175 x6 − 129594465x4 − 1166336325 x2

)I1(x)− 1166336640Ψ(x)

n = 10

256∫

P10(x) · J0(x) dx =∫ (

46189 x10 − 109395 x8 + 90090 x6 − 30030 x4 + 3465 x2 − 63)

J0(x) dx =

=(415701 x9 − 26954928 x7 + 943872930x5 − 14158184040x3 − 63 x

)J0(x)+

+(46189 x10 − 3850704 x8 + 188774586x6 − 4719394680x4 + 42474555585x2

)J1(x)− 42474555648 Φ(x)

256∫

P10(x) · J1(x) dx =

=(−46189 x10 + 3804515x8 − 182706810x6 + 4384993470 x4 − 35079951225x2 + 63

)J0(x)+

+(461890 x9 − 30436120 x7 + 1096240860 x5 − 17539973880x3 + 70159902450x

)J1(x)

256∫

P10(x) · I0(x) dx =(−415701 x9 − 25423398 x7 − 890269380x5 − 13353950610x3 − 63 x

)I0(x)+

+(46189 x10 + 3631914x8 + 178053876x6 + 4451316870 x4 + 40061855295x2

)I1(x) + 40061855232 Ψ(x)

256∫

P10(x)·I1(x) dx =(46189 x10 + 3585725 x8 + 172204890x6 + 4132887330 x4 + 33063102105x2 − 63

)I0(x)+

+(−461890 x9 − 28685800 x7 − 1033229340 x5 − 16531549320x3 − 66126204210x) I1(x)

n = 11

256∫

P11(x) · J0(x) dx =

=∫ (

88179 x11 − 230945 x9 + 218790 x7 − 90090 x5 + 15015 x3 − 693 x)

J0(x) dx =

=(881790 x10 − 72390760 x8 + 3476069220x6 − 83426021640x4 + 667408203150x2

)J0(x)+

+(88179 x11 − 9048845x9 + 579344870x7 − 20856505410 x5 + 333704101575x3 − 1334816406993x

)J1(x)

256∫

P11(x) · J1(x) dx =

=(−88179 x11 + 8960666 x9 − 564740748 x7 + 19766016270x5 − 296490259065 x3

)J0(x)+

+(969969 x10 − 80645994 x8 + 3953185236x6 − 98830081350x4 + 889470777195x2

)J1(x)−

−889470777888Φ(x)

256∫

P11(x) · I0(x) dx =

=(−881790 x10 − 68695640 x8 − 3298703460x6 − 79168522680x4 − 633348211470x2

)I0(x)+

+(88179 x11 + 8586955 x9 + 549783910x7 + 19792130670x5 + 316674105735x3 + 1266696422247x

)I1(x)

256∫

P11(x) · I1(x) dx =

=(88179 x11 + 8498776 x9 + 535641678x7 + 18747368640x5 + 281210544615x3

)I0(x)+

+(−969969 x10 − 76488984 x8 − 3749491746 x6 − 93736843200x4 − 843631633845x2

)I1(x)−

−843631633152Ψ(x)

n = 12

1024∫

P12(x) · J0(x) dx =

121

=∫ (

676039 x12 − 1939938 x10 + 2078505x8 − 1021020 x6 + 225225 x4 − 18018 x2 + 231)

J0(x) dx =

=(7436429x11 − 753665913 x9 + 47495502054x7 − 1662347676990x5 + 24935215830525x3 + 231 x

)J0(x)+

+(676039x12 − 83740657 x10 + 6785071722 x8 − 332469535398x6 + 8311738610175x4−

−74805647509593x2) J1(x) + 74805647509824Φ(x)

1024∫

P12(x) · J1(x) dx = (−676039 x12 + 83064618x10 − 6647247945 x8 + 319068922380x6−

−7657654362345x4 + 61261234916778x2 − 231) J0(x) + (8112468x11 − 830646180x9 + 53177983560x7−

−1914413534280x5 + 30630617449380x3 − 122522469833556x) J1(x)

1024∫

P12(x) · I0(x) dx = (−7436429 x11 − 718747029x9 − 45295612362x7 − 1585341327570x5−

−23780120589225x3 + 231 x) I0(x) + (676039 x12 + 79860781 x10 + 6470801766x8 + 317068265514x6+

+7926706863075x4 + 71340361749657x2) I1(x) + 71340361749888Ψ(x)

1024∫

P12(x)·I1(x) dx = (676039 x12+79184742x10+6336857865x8+304168156500x6+7300035981225x4+

+58400287831782x2 + 231) I0(x) + (−8112468x11 − 791847420x9 − 50694862920x7 − 1825008939000x5−

−29200143924900x3 − 116800575663564x) I1(x)

n = 13

1024∫

P13(x) · J0(x) dx =

=∫ (

1300075 x13 − 4056234x11 + 4849845x9 − 2771340 x7 + 765765 x5 − 90090 x3 + 3003 x)

J0(x) dx =

= (15600900 x12 − 1912670340 x10 + 153052425960x8 − 7346533074120x6 + 176316796841940x4−

−1410534374915700x2) J0(x) + (1300075x13 − 191267034x11 + 19131553245x9 − 1224422179020x7+

+44079199210485x5 − 705267187457850x3 + 2821068749834403x) J1(x)

1024∫

P13(x) · J1(x) dx = (−1300075 x13 + 189966959x11 − 18811578786x9 + 1185132234858x7−

−41479628985795x5 + 622194434877015x3) J0(x) + (16900975 x12 − 2089636549x10 + 169304209074x8−

−8295925644006x6 + 207398144928975x4 − 1866583304631045x2) J1(x) + 1866583304634048 Φ(x)∫P13(x) · I0(x) dx = (−15600900 x12 − 1831545660 x10 − 146562451560x8 − 7034981046840x6−

−168839548187220x4 − 1350716385317580x2) I0(x) + (1300075x13 + 183154566x11 + 18320306445x9+

+1172496841140x7 + 42209887046805x5 + 675358192658790x3 + 2701432770638163x) I1(x)

1024∫

P13(x) · I1(x) dx = (1300075 x13 + 181854491x11 + 18008444454x9 + 1134529229262x7+

+39708523789935x5 +595627856758935 x3) I0(x)+ (−16900975 x12− 2000399401x10− 162076000086 x8−

−7941704604834x6 − 198542618949675x4 − 1786883570276805x2) I1(x)− 1786883570279808Ψ(x)

n = 14

2048∫

P14(x) · J0(x) dx =∫

(5014575x14 − 16900975 x12 + 22309287 x10 − 14549535 x8 + 4849845x6−

−765765 x4 + 45045 x2 − 429) J0(x) dx = (65189475 x13 − 9508005650 x11 + 941493342933x9−

−59314182451524x7 + 2075996410052565x5 − 31139946153085770x3 − 429 x) J0(x)+

+(5014575x14 − 864364150x12 + 104610371437x10 − 8473454635932x8 + 415199282010513x6−

122

−10379982051028590x4 + 93419838459302355x2) J1(x)− 93419838459302784Φ(x)

2048∫

P14(x) · J1(x) dx = (−5014575 x14 + 859349575x12 − 103144258287x10 + 8251555212495x8−

−396074655049605x6 + 9505791721956285x4 − 76046333775695325x2 + 429) J0(x)+

+(70204050 x13 − 10312194900x11 + 1031442582870x9 − 66012441699960x7 + 2376447930297630x5−

−38023166887825140x3 + 152092667551390650x) J1(x)

2048∫

P14(x) · I0(x) dx = (−65189475 x13 − 9136184200 x11 − 904683019383x9 − 56994928374384x7−

−1994822517352665x5 − 29922337757992680x3 − 429 x) I0(x) + (5014575x14 + 830562200x12+

+100520335487x10 + 8142132624912x8 + 398964503470533x6 + 9974112585997560x4+

+89767013274023085x2) I1(x) + 89767013274022656Ψ(x)

2048∫

P14(x) · I1(x) dx = (5014575 x14 + 825547625x12 + 99088024287x10 + 7927027393425x8+

+380497319734245x6 + 9131935672856115x4 + 73055485382893965x2 − 429) I0(x)+

+(−70204050 x13 −−9906571500 x11 − 990880242870x9 − 63416219147400x7 − 2282983918405470x5−

−36527742691424460x3 − 146110970765787930x) I1(x)

n = 15

2048∫

P15(x) · J0(x) dx =∫

(9694845x15 − 35102025 x13 + 50702925 x11 − 37182145 x9 + +14549535x7−

−2909907x5 + 255255 x3 − 6435 x) J0(x) dx = (135727830 x14 − 23223499740x12 + 2787326998050x10−

−222986457301160x8+10703350037752890x6−256880400917708988x4+2055043207342182414x2) J0(x)+

+(9694845x15 − 1935291645 x13 + 278732699805x11 − 27873307162645x9 + 1783891672958815x7−

−64220100229427247x5 + 1027521603671091207x3 − 4110086414684371263x) J1(x)

2048∫

P15(x) · J1(x) dx = (−9694845 x15 + 1925596800 x13 − 275411045325x11 + 27265730669320x9−

−1717741046716695x7 + 60120936637994232x5 − 901814049570168735x3) J0(x) + (145422675x14−

−25032758400x12 + 3029521498575x10 − 245391576023880x8 + 12024187327016865x6−

−300604683189971160x4 + 2705442148710506205x2) J1(x)− 2705442148710512640Φ(x)∫P15(x) · I0(x) dx = (−135727830x14 − 22381051140x12 − 2686233166050x10 − 214898355826840x8−

−10315121166985530x6 − 247562907996013092x4 − 1980503263968615246x2) I0(x)+

+(9694845x15 + 1865087595 x13 + 268623316605x11 + 26862294478355x9 + 1719186861164255x7+

−61890726999003273x5 + 990251631984307623x3 + 3961006527937224057x) I1(x)

2048∫

P15(x) · I1(x) dx = (9694845 x15 + 1855392750x13 + 265371866175x11 + 26271777569180x9+

+1655122001407875x7 + 57929270046365718x5 + 868939050695741025x3) I0(x)+

+(−145422675x14−24120105750x12−2919090527925x10−236445998122620x8−11585854009855125x6−

−289646350231828590x4 − 2606817152087223075x2) I1(x)− 2606817152087216640Ψ(x)

123

a) Chebyshev Polynomials of the first kind Tn(x) :

Weight function: w(x) = (1− x2)−1/2, interval −1 ≤ x ≤ 1.

Holds∫

T0(x) Zν(x) dx =∫

Zν(x) dx and∫

T1(x) Zν(x) dx =∫

xZν(x) dx .

n = 2 ∫T2(x) · J0(x) dx =

∫(2 x2 − 1) J0(x) dx = −xJ0(x) + 2x2 J1(x)− 3 Φ(x)∫

T2(x) · J1(x) dx = (−2 x2 + 1) J0(x) + 4xJ1(x)∫T2(x) · I0(x) dx = −x I0(x) + 2x2 I1(x) + Ψ(x)∫

T2(x) · I1(x) dx = (2x2 − 1) I0(x)− 4 I1(x)

n = 3 ∫T3(x) · J0(x) dx =

∫(4 x3 − 3 x) J0(x) dx = 8x2 J0(x) + (4 x3 − 19 x) J1(x)∫

T3(x) · J1(x) dx = −4x3 J0(x) + 12x2 J1(x)− 15 Φ(x)∫T3(x) · I0(x) dx = −8 x2 I0(x) + (4 x3 + 13 x) I1(x)∫T3(x) · I1(x) dx = 4x3 I0(x)− 12x2 I1(x)− 9 Ψ(x)

n = 4∫T4(x) · J0(x) dx =

∫(8 x4 − 8 x2 + 1) J0(x) dx = (24 x3 + x) J0(x) + (8 x4 − 80 x2) J1(x) + 81 Φ(x)

∫T4(x) · J1(x) dx = (−8 x4 + 72 x2 − 1) J0(x) + (32 x3 − 144 x) J1(x)∫T4(x) · I0(x) dx = (−24x3 + x) I0(x) + (8 x4 + 64 x2) I1(x) + 65 Ψ(x)∫T4(x) · I1(x) dx = (8 x4 + 56 x2 + 1) I0(x) + (−32 x3 − 112 x) I1(x)

n = 5 ∫T5(x) · J0(x) dx =

∫(16 x5 − 20 x3 + 5 x) J0(x) dx =

= (64 x4 − 552 x2) J0(x) + (16 x5 − 276 x3 + 1109 x) J1(x)∫T5(x) · J1(x) dx = (−16 x5 + 260 x3) J0(x) + (80 x4 − 780 x2) J1(x) + 785 Φ(x)∫

T5(x) · I0(x) dx = (−64 x4 − 472 x2) I0(x) + (16 x5 + 236 x3 + 949 x) I1(x)∫T5(x) · I1(x) dx = (16 x5 + 220 x3) I0(x) + (−80 x4 − 660 x2) I1(x)− 665 Ψ(x)

n = 6 ∫T6(x) · J0(x) dx =

∫(32 x6 − 48 x4 + 18 x2 − 1) J0(x) dx =

= (160 x5 − 2544 x3 − x) J0(x) + (32 x6 − 848 x4 + 7650 x2) J1(x)− 7651 Φ(x)∫T6(x) · J1(x) dx = (−32 x6 + 816 x4 − 6546 x2 + 1) J0(x) + (192x5 − 3264 x3 + 13092 x) J1(x)∫

T6(x) · I0(x) dx = (−160 x5 − 2256 x3 − x) I0(x) + (32 x6 + 752 x4 + 6786 x2) I1(x) + 6785 Ψ(x)

124

∫T6(x) · I1(x) dx = (32 x6 + 720 x4 + 5778 x2 − 1) I0(x) + (−192 x5 − 2880 x3 − 11556 x) I1(x)

n = 7 ∫T7(x) · J0(x) dx =

∫(64 x7 − 112 x5 + 56 x3 − 7 x) J0(x) dx =

= (384 x6 − 9664 x4 + 77424 x2) J0(x) + (64 x7 − 2416 x5 + 38712 x3 − 154855 x) J1(x)∫T7(x)·J1(x) dx = (−64 x7+2352 x5−35336 x3) J0(x)+(448 x6−11760 x4+106008 x2) J1(x)−106015 Φ(x)∫

T7(x) · I0(x) dx =

= (−384 x6 − 8768 x4 − 70256 x2) I0(x) + (64 x7 + 2192 x5 + 35128 x3 + 140505 x) I1(x)∫T7(x) ·I1(x) dx = (64 x7 +2128 x5 +31976 x3) I0(x)+(−448 x6−10640 x4−95928 x2) I1(x)−95921 Ψ(x)

n = 8 ∫T8(x) · J0(x) dx =

∫(128x8 − 256 x6 + 160 x4 − 32 x2 + 1) J0(x) dx =

= (896 x7−32640 x5+490080 x3+x)J0(x)+(128x8−6528 x6+163360 x4−1470272 x2)J1(x)+1470273Φ(x)∫T8(x) · J1(x) dx = (−128 x8 + 6400 x6 − 153760 x4 + 1230112x2 − 1) J0(x)+

+(1024x7 − 38400 x5 + 615040 x3 − 2460224x) J1(x)∫T8(x) · I0(x) dx = (−896 x7 − 30080 x5 − 451680 x3 + x) I0(x)+

+(128x8 + 6016 x6 + 150560 x4 + 1355008x2) I1(x) + 1355009 Ψ(x)∫T8(x) · I1(x) dx = (128 x8 + 5888 x6 + 141472 x4 + 1131744x2 + 1) I0(x)+

+(−1024 x7 − 35328 x5 − 565888 x3 − 2263488 x) I1(x)

n = 9 ∫T9(x) · J0(x) dx =

∫(256 x9 − 576 x7 + 432 x5 − 120 x3 + 9 x) J0(x) dx =

= (2048 x8 − 101760 x6 + 2443968x4 − 19551984 x2) J0(x)+

+(256x9 − 16960 x7 + 610992 x5 − 9775992 x3 + 39103977) J1(x)∫T9(x) · J1(x) dx = (−256 x9 + 16704 x7 − 585072 x5 + 8776200 x3) J0(x)+

+(2304x8 − 116928 x6 + 2925360x4 − 26328600 x2) J1(x) + 26328609 Φ(x)∫T9(x) · I0(x) dx = (−2048 x8 − 94848 x6 − 2278080 x4 − 18224400 x2) I0(x)+

+(256x9 + 15808x7 + 569520 x5 + 9112200x3 + 36448809 x) I1(x)∫T9(x) · I1(x) dx = (256 x9 + 15552 x7 + 544752 x5 + 8171160x3) I0(x)+

+(−2304 x8 − 108864 x6 − 2723760x4 − 24513480 x2) I1(x)− 24513489 Ψ(x)

n = 10 ∫T10(x) · J0(x) dx =

∫(512 x10 − 1280 x8 + 1120 x6 − 400 x4 + 50 x2 − 1) J0(x) dx =

= (4608 x9 − 299264 x7 + 10479840 x5 − 157198800x3 − x) J0(x)+

+(512x10 − 42752 x8 + 2095968 x6 − 52399600 x4 + 471596450x2) J1(x)− 471596451Φ(x)

125

∫T10(x) · J1(x) dx = (−512 x10 + 42240 x8 − 2028640x6 + 48687760x4 − 389502130x2 + 1) J0(x)+

+(5120x9 − 337920 x7 + 12171840 x5 − 194751040 x3 + 779004260x) J1(x)∫T10(x) · I0(x) dx = (−4608 x9 − 281344 x7 − 9852640x5 − 147788400 x3 − x) I0(x)+

+(512x10 + 40192 x8 + 1970528 x6 + 49262800x4 + 443365250x2) I1(x) + 443365249 Ψ(x)∫T10(x) · I1(x) dx = (512 x10 + 39680 x8 + 1905760x6 + 45737840 x4 + 365902770x2 − 1) I0(x)+

+(−5120 x9 − 317440 x7 − 11434560 x5 − 182951360 x3 − 731805540x) I1(x)

n = 11∫T11(x) · J0(x) dx =

∫(1024 x11 − 2816 x9 + 2816 x7 − 1232 x5 + 220 x3 − 11 x) J0(x) dx =

= (10240 x10 − 841728 x8 + 40419840 x6 − 970081088x4 + 7760649144 x2) J0(x)+

+(1024x11 − 105216 x9 + 6736640 x7 − 242520272 x5 + 3880324572x3 − 15521298299x) J1(x)∫T11(x) · J1(x) dx = (−1024 x11 + 104192 x9 − 6566912x7 + 229843152x5 − 3447647500 x3) J0(x)+

+(11264x10 − 937728 x8 + 45968384 x6 − 1149215760 x4 + 10342942500x2) J1(x)− 10342942511Φ(x)∫T11(x) · I0(x) dx = (−10240 x10 − 796672 x8 − 38257152 x6 − 918166720 x4 − 7345334200x2) I0(x)+

+(1024x11 + 99584 x9 + 6376192x7 + 229541680 x5 + 3672667100x3 + 14690668389x) I1(x)∫T11(x) · I1(x) dx = (1024 x11 + 98560 x9 + 6212096x7 + 217422128x5 + 3261332140x3) I0(x)+

+(−11264 x10 − 887040 x8 − 43484672 x6 − 1087110640 x4 − 9783996420 x2) I1(x)− 9783996409 Ψ(x)

n = 12∫T12(x) · J0(x) dx =

∫(2048x12 − 6144 x10 + 6912 x8 − 3584 x6 + 840 x4 − 72 x2 + 1) J0(x) dx =

= (22528 x11 − 2285568x9 + 144039168x7 − 5041388800 x5 + 75620834520x3 + x) J0(x)+

+(2048x12 − 253952 x10 + 20577024 x8 − 1008277760 x6 + 25206944840x4 − 226862503632x2) J1(x)+

+226862503633Φ(x)∫T12(x) · J1(x) dx = (−2048 x12 + 251904 x10 − 20159232 x8 + 967646720x6 − 23223522120x4+

+185788177032x2 − 1) J0(x) + (24576 x11 − 2519040 x9 + 161273856x7 − 5805880320x5+

+92894088480x3 − 371576354064 x) J1(x)∫T12(x)·I0(x) dx = (−22528 x11−2174976x9−137071872x7−4797497600 x5−71962466520x3+x) I0(x)+

+(2048x12 + 241664 x10 + 19581696 x8 + 959499520x6 + 23987488840x4 + 215887399488x2) I1(x)+

+215887399489Ψ(x)∫T12(x) · I1(x) dx =

= (2048 x12 + 239616 x10 + 19176192 x8 + 920453632x6 + 22090888008x4 + 176727103992x2 + 1) I0(x)+

+(−24576 x11 − 2396160 x9 − 153409536x7 − 5522721792 x5 − 88363552032x3 − 353454207984 x) I1(x)

126

n = 13 ∫T13(x) · J0(x) dx =

=∫

(4096x13 − 13312 x11 + 16640x9 − 9984 x7 + 2912 x5 − 364 x3 + 13 x) J0(x) dx =

= (49152 x12−6031360 x10+482641920 x8−23166872064x6+556004941184 x4−4448039530200x2) J0(x)+

+(4096x13 − 603136 x11 + 60330240 x9 − 3861145344x7 + 139001235296x5 − 2224019765100x3+

+8896079060413x) J1(x)∫T13(x) · J1(x) dx = (−4096 x13 + 599040 x11 − 59321600 x9 + 3737270784 x7 − 130804480352 x5+

+1962067205644x3) J0(x)+(53248 x12−6589440x10+533894400 x8−26160895488x6+654022401760 x4−

−5886201616932x2) J1(x) + 5886201616945 Φ(x)∫T13(x) · I0(x) dx = (−49152 x12 − 5765120 x10 − 461342720x8 − 22144390656x6−

−531465387392x4 − 4251723098408x2) I0(x) + (4096 x13 + 576512 x11 + 57667840 x9+

+3690731776x7 + 132866346848x5 + 2125861549204x3 + 8503446196829x) I1(x)∫T13(x) · I1(x) dx = (4096 x13 + 572416 x11 + 56685824x9 + 3571196928 x7 + 124991895392x5+

+1874878430516x3) I0(x) + (−53248 x12 − 6296576 x10 − 510172416x8 − 24998378496x6−

−624959476960 x4 − 5624635291548x2) I1(x)− 5624635291561Ψ(x)

n = 14∫T14(x)·J0(x) dx =

∫(8192x14−28672 x12+39424 x10−26880 x8+9408 x6−1568 x4+98x2−1) J0(x) dx =

= (106496 x13−15544320 x11 +1539242496 x9−96972465408x7 +3394036336320 x5−50910545049504 x3−

−x) J0(x)+(8192 x14−1413120 x12+171026944x10−13853209344x8+678807267264x6−16970181683168x4+

+152731635148610x2) J1(x)− 152731635148611Φ(x)∫T14(x) · J1(x) dx = (−8192 x14 + 1404928 x12 − 168630784x10 + 13490489600x8 − 647543510208x6+

+15541044246560x4 − 124328353972578x2 + 1) J0(x) + (114688 x13 − 16859136 x11 + 1686307840x9−

−107923916800x7 + 3885261061248x5 − 62164176986240x3 + 248656707945156x) J1(x)∫T14(x) ·I0(x) dx = (−106496 x13−14913536 x11−1476794880x9−93037889280x7−3256326171840x5−

−48844892572896x3−x) I0(x)+(8192 x14+1355776x12+164088320x10+13291127040x8+651265234368x6+

+16281630857632x4 + 146534677718786x2) I1(x) + 146534677718785Ψ(x)∫T14(x) · I1(x) dx = (8192 x14 + 1347584x12 + 161749504x10 + 12939933440x8 + 621116814528x6+

+14906803547104x4 + 119254428376930x2 − 1) I0(x) + (−114688 x13 − 16171008 x11 − 1617495040x9−

−103519467520 x7 − 3726700887168x5 − 59627214188416x3 − 238508856753860x) I1(x)

n = 15 ∫T15(x) · J0(x) dx =

=∫

(16384x15 − 61440 x13 + 92160x11 − 70400 x9 + 28800x7 − 6048 x5 + 560 x3 − 15 x) J0(x) dx =

= (229376 x14 − 39272448 x12 + 4713615360x10 − 377089792000x8 + 18100310188800x6−

127

−434407444555392x4 + 3475259556444256x2) J0(x) + (16384x15 − 3272704 x13 + 471361536x11−

−47136224000x9+3016718364800x7−108601861138848x5+1737629778222128 x3−6950519112888527x) J1(x)∫T15(x) · J1(x) dx = (−16384 x15 + 3256320x13 − 465745920x11 + 46108916480x9 − 2904861767040x7+

+101670161852448x5 − 1525052427787280x3) J0(x) + (245760 x14 − 42332160 x12 + 5123205120x10−

−414980248320x8 + 20334032369280x6 − 508350809262240x4 + 4575157283361840x2) J1(x)−

−4575157283361855Φ(x)∫T15(x) · I0(x) dx = (−229376 x14 − 37797888 x12 − 4536668160 x10 − 362932889600x8−

−17420778873600x6 − 418098692942208x4 − 3344789543538784x2) I0(x)+

+(16384x15 + 3149824x13 + 453666816x11 + 45366611200x9 + 2903463145600x7+

+104524673235552x5 + 1672394771769392x3 + 6689579087077553x) I1(x)∫T15(x) · I1(x) dx = (16384 x15 + 3133440x13 + 448174080x11 + 44369163520x9 + 2795257330560x7+

+97834006563552x5 + 1467510098453840x3) I0(x) + (−245760 x14 − 40734720 x12 − 4929914880 x10−

−399322471680 x8 − 19566801313920x6 − 489170032817760x4 − 4402530295361520x2) I1(x)−

−4402530295361505Ψ(x)

128

a) Chebyshev Polynomials of the second kind Un(x) :

Weight function: w(x) = (1− x2)1/2, interval −1 ≤ x ≤ 1.

Holds∫

U0(x)Zν(x) dx =∫

Zν(x) dx and∫

U1(x) Zν(x) dx =∫

2xZν(x) dx .

n = 2 ∫U2(x) · J0(x) dx =

∫(4x2 − 1) J0(x) dx = −xJ0(x) + 4x2 J1(x)− 5 Φ(x)∫

U2(x) · J1(x) dx = (−4x2 + 1) J0(x) + 8xJ1(x)∫U2(x) · I0(x) dx = −x I0(x) + 4x2 I1(x) + 3 Ψ(x)∫

U2(x) · I1(x) dx = (4x2 − 1) I0(x)− 8x I1(x)

n = 3 ∫U3(x) · J0(x) dx =

∫(8 x3 − 4 x) J0(x) dx = 16x2 J0(x) + (8x3 − 36x) J1(x)∫

U3(x) · J1(x) dx = −8x3 J0(x) + 24x2 J1(x)− 28 Φ(x)∫U3(x) · I0(x) dx = −16x2 I0(x) + (8x3 + 28x) I1(x)∫U3(x) · I1(x) dx = 8x3 I0(x)− 24x2 I1(x)− 20 Ψ(x)

n = 4∫U4(x) · J0(x) dx =

∫(16 x4 − 12 x2 + 1) J0(x) dx = (48 x3 + x) J0(x) + (16 x4 − 156 x2) J1(x) + 157 Φ(x)

∫U4(x) · J1(x) dx = (−16 x4 + 140 x2 − 1) J0(x) + (64 x3 − 280 x) J1(x)∫

U4(x) · I0(x) dx = (−48 x3 + x) I0(x) + (16 x4 + 132 x2) I1(x) + 133 Ψ(x)∫U4(x) · I1(x) dx = (16 x4 + 116 x2 + 1) I0(x) + (−64 x3 − 232 x) I1(x)

n = 5∫U5(x)·J0(x) dx =

∫(32 x5−32 x3+6x) J0(x) dx = (128 x4−1088 x2) J0(x)+(32x5−544 x3+2182x) J1(x)

∫U5(x) · J1(x) dx = (−32 x5 + 512 x3) J0(x) + (160x4 − 1536 x2) J1(x) + 1542 Φ(x)∫

U5(x) · I0(x) dx = (−128 x4 − 960 x2) I0(x) + (32 x5 + 480 x3 + 1926 x) I1(x)∫U5(x) · I1(x) dx = (32 x5 + 448 x3) I0(x) + (−160 x4 − 1344 x2) I1(x)− 1350 Ψ(x)

n = 6 ∫U6(x) · J0(x) dx =

∫(64 x6 − 80 x4 + 24 x2 − 1) J0(x) dx =

= (320 x5 − 5040 x3 − x) J0(x) + (64 x6 − 1680 x4 + 15144x2) J1(x)− 15145 Φ(x)∫U6(x) · J1(x) dx = (−64 x6 + 1616 x4 − 12952 x2 + 1) J0(x) + (384 x5 − 6464 x3 + 25904 x) J1(x)∫

U6(x) · I0(x) dx = (−320 x5 − 4560 x3 − x) I0(x) + (64 x6 + 1520 x4 + 13704x2) I1(x) + 13703 Ψ(x)

129

∫U6(x) · I1(x) dx = (64 x6 + 1456 x4 + 11672 x2 − 1) I0(x) + (−384 x5 − 5824 x3 − 23344 x) I1(x)

n = 7 ∫U7(x) · J0(x) dx =

∫(128x7 − 192 x5 + 80 x3 − 8 x) J0(x) dx =

= (768 x6 − 19200 x4 + 153760 x2) J0(x) + (128x7 − 4800 x5 + 76880 x3 − 307528 x) J1(x)∫U7(x)·J1(x) dx = (−128 x7+4672x5−70160 x3) J0(x)+(896 x6−23360 x4+210480x2) J1(x)−210488 Φ(x)∫U7(x) ·I0(x) dx = (−768 x6−17664 x4−141472 x2) I0(x)+(128 x7 +4416 x5 +70736 x3 +282936 x) I1(x)∫U7(x)·I1(x) dx = (128 x7+4288 x5+64400 x3) I0(x)+(−896 x6−21440 x4−193200 x2) I1(x)−193192 Ψ(x)

n = 8 ∫U8(x) · J0(x) dx =

∫(256 x8 − 448 x6 + 240 x4 − 40 x2 + 1) J0(x) dx =

= (1792 x7−64960 x5+975120x3+x) J0(x)+(256 x8−12992 x6+325040x4−2925400 x2) J1(x)+2925401Φ(x)∫U8(x) · J1(x) dx = (−256 x8 + 12736x6 − 305904 x4 + 2447272x2 − 1) J0(x)+

+(2048x7 − 76416 x5 + 1223616 x3 − 4894544 x) J1(x)∫U8(x) · I0(x) dx = (−1792 x7 − 60480 x5 − 907920 x3 + x) I0(x)+

+(256x8 + 12096 x6 + 302640 x4 + 2723720 x2) I1(x) + 2723721 Ψ(x)∫U8(x) · I1(x) dx = (256 x8 + 11840 x6 + 284400 x4 + 2275160x2 + 1) I0(x)+

+(−2048 x7 − 71040 x5 − 1137600x3 − 4550320 x) I1(x)

n = 9 ∫U9(x) · J0(x) dx =

∫(512x9 − 1024 x7 + 672 x5 − 160 x3 + 10 x) J0(x) dx =

= (4096 x8 − 202752 x6 + 4868736x4 − 38950208 x2) J0(x)+

+(512x9 − 33792 x7 + 1217184x5 − 19475104 x3 + 77900426 x) J1(x)∫U9(x) · J1(x) dx = (−512 x9 + 33280 x7 − 1165472x5 + 17482240x3) J0(x)+

+(4608x8 − 232960 x6 + 5827360x4 − 52446720 x2) J1(x) + 52446730Φ(x)∫U9(x) · I0(x) dx = (−4096 x8 − 190464 x6 − 4573824 x4 − 36590272 x2) I0(x)+

+(512x9 + 31744 x7 + 1143456x5 + 18295136x3 + 73180554 x) I1(x)∫U9(x) · I1(x) dx = (512 x9 + 31232x7 + 1093792x5 + 16406720 x3) I0(x)+

+(−4608 x8 − 218624 x6 − 5468960x4 − 49220160 x2) I1(x)− 49220170 Ψ(x)

n = 10 ∫U10(x) · J0(x) dx =

∫(1024x10 − 2304 x8 + 1792 x6 − 560 x4 + 60 x2 − 1) J0(x) dx =

= (9216 x9 − 596736 x7 + 20894720 x5 − 313422480x3 − x) J0(x)+

+(1024x10 − 85248 x8 + 4178944 x6 − 104474160x4 + 940267500x2) J1(x)− 940267501 Φ(x)∫U10(x) · J1(x) dx = (−1024 x10 + 84224 x8 − 4044544 x6 + 97069616 x4 − 776556988x2 + 1) J0(x)+

130

+(10240x9 − 673792 x7 + 24267264 x5 − 388278464 x3 + 1553113976x) J1(x)∫U10(x) · I0(x) dx = (−9216 x9 − 564480 x7 − 19765760 x5 − 296484720x3 − x) I0(x)+

+(1024x10 + 80640 x8 + 3953152x6 + 98828240 x4 + 889454220x2) I1(x) + 889454219 Ψ(x)∫U10(x) · I1(x) dx = (1024 x10 + 79616x8 + 3823360 x6 + 91760080 x4 + 734080700x2 − 1) I0(x)+

+(−10240 x9 − 636928 x7 − 22940160 x5 − 367040320 x3 − 1468161400 x) I1(x)

131

a) Laguerre Polynomials Ln(x) :

Weight function: w(x) = e−x, interval 0 ≤ x < ∞Holds

∫L0(x) Zν(x) dx =

∫Zν(x) dx .

n = 1 ∫L1(x) · J0(x) dx =

∫(−x + 1) J0(x) dx = xJ0(x)− xJ1(x) + Φ(x)∫

L1(x) · J1(x) dx = −J0(x)− Φ(x)∫L1(x) · I0(x) dx = x I0(x)− x I1(x) + Ψ(x)∫

L1(x) · I1(x) dx = I0(x) + Ψ(x)

n = 2 ∫L2(x) · J0(x) dx =

∫(x2 − 4 x + 2) J0(x) dx = 2xJ0(x) + (x2 − 4x) J1(x) + Φ(x)∫

L2(x) · J1(x) dx = (−x2 − 2) J0(x) + 2xJ1(x)− 4 Φ(x)∫L2(x) · I0(x) dx = 2x I0(x) + (x2 − 4x) I1(x) + 3 Ψ(x)∫L2(x) · I1(x) dx = (x2 + 2) I0(x)− 2x I1(x) + 4 Ψ(x)

n = 3∫L3(x)·J0(x) dx =

∫(−x3+9 x2−18 x+6) J0(x) dx = (−2 x2+6 x) J0(x)+(−x3+9 x2−14 x) J1(x)−3 Φ(x)

∫L3(x) · J1(x) dx = (x3 − 9 x2 − 6) J0(x) + (−3 x2 + 18 x) J1(x)− 15 Φ(x)∫L3(x) · I0(x) dx = (2 x2 + 6 x) I0(x) + (−x3 + 9 x2 − 22 x) I1(x) + 15 Ψ(x)∫L3(x) · I1(x) dx = (−x3 + 9 x2 + 6) I0(x) + (3 x2 − 18 x) I1(x) + 21 Ψ(x)

n = 4 ∫L4(x) · J0(x) dx =

∫(x4 − 16 x3 + 72 x2 − 96 x + 24) J0(x) dx =

= (3 x3 − 32 x2 + 24 x) J0(x) + (x4 − 16 x3 + 63 x2 − 32 x) J1(x)− 39 Φ(x)∫L4(x) · J1(x) dx = (−x4 + 16 x3 − 64 x2 − 24) J0(x) + (4 x3 − 48 x2 + 128 x) J1(x)− 48 Φ(x)∫

L4(x) · I0(x) dx = (−3 x3 + 32 x2 + 24 x) I0(x) + (x4 − 16 x3 + 81 x2 − 160 x) I1(x) + 105 Ψ(x)∫L4(x) · I1(x) dx = (x4 − 16 x3 + 80 x2 + 24) I0(x) + (−4 x3 + 48 x2 − 160 x) I1(x) + 144 Ψ(x)

n = 5 ∫L5(x) · J0(x) dx =

∫(−x5 + 25 x4 − 200 x3 + 600 x2 − 600 x + 120) J0(x) dx =

= (−4 x4 + 75 x3 − 368 x2 + 120 x) J0(x) + (−x5 + 25 x4 − 184 x3 + 375 x2 + 136 x) J1(x)− 255 Φ(x)∫L5(x)·J1(x) dx = (x5−25 x4+185x3−400 x2−120) J0(x)+(−5 x4+100x3−555 x2+800x) J1(x)−45 Φ(x)∫L5(x)·I0(x) dx = (4 x4−75 x3+432x2+120x) I0(x)+(−x5+25x4−216 x3+825x2−1464 x) I1(x)+945Ψ(x)

132

∫L5(x)·I1(x) dx = (−x5+25x4−215 x3+800x2+120) I0(x)+(5x4−100 x3+645x2−1600 x) I1(x)+1245Ψ(x)

n = 6 ∫L6(x) · J0(x) dx =

∫(x6 − 36 x5 + 450 x4 − 2400 x3 + 5400 x2 − 4320 x + 720) J0(x) dx =

= (5 x5 − 144 x4 + 1275 x3 − 3648 x2 + 720 x) J0(x)+

+(x6 − 36 x5 + 425 x4 − 1824 x3 + 1575 x2 + 2976 x) J1(x)− 855 Φ(x)∫L6(x) · J1(x) dx = (−x6 + 36 x5 − 426 x4 + 1860 x3 − 1992 x2 − 720) J0(x)+

+(6x5 − 180 x4 + 1704 x3 − 5580 x2 + 3984 x) J1(x) + 1260 Φ(x)∫L6(x) · I0(x) dx = (−5 x5 + 144 x4 − 1425 x3 + 5952 x2 + 720 x) I0(x)+

+(x6 − 36 x5 + 475 x4 − 2976 x3 + 9675 x2 − 16224 x) I1(x) + 10395 Ψ(x)∫L6(x) · I1(x) dx = (x6 − 36 x5 + 474 x4 − 2940 x3 + 9192 x2 + 720) I0(x)+

+(−6 x5 + 180 x4 − 1896 x3 + 8820 x2 − 18384 x) I1(x) + 13140 Ψ(x)

n = 7∫L7(x) · J0(x) dx =

∫(−x7 + 49 x6− 882 x5 + 7350 x4− 29400 x3 + 52920 x2− 35280 x + 5040) J0(x) dx =

= (−6 x6 + 245 x5 − 3384 x4 + 18375 x3 − 31728 x2 + 5040 x) J0(x)+

+(−x7 + 49 x6 − 846 x5 + 6125 x4 − 15864 x3 − 2205 x2 + 28176 x) J1(x) + 7245Φ(x)∫L7(x) · J1(x) dx = (x7 − 49 x6 + 847 x5 − 6174 x4 + 16695 x3 − 3528 x2 − 5040) J0(x)+

+(−7 x6 + 294 x5 − 4235 x4 + 24696 x3 − 50085 x2 + 7056 x) J1(x) + 14805 Φ(x)∫L7(x) · I0(x) dx = (6 x6 − 245 x5 + 3672 x4 − 25725 x3 + 88176x2 + 5040 x) I0(x)+

+(−x7 + 49 x6 − 918 x5 + 8575 x4 − 44088 x3 + 130095 x2 − 211632 x) I1(x) + 135135 Ψ(x)∫L7(x) · I1(x) dx = (−x7 + 49 x6 − 917 x5 + 8526 x4 − 43155 x3 + 121128 x2 + 5040) I0(x)+

+(7x6 − 294 x5 + 4585 x4 − 34104 x3 + 129465 x2 − 242256 x) I1(x) + 164745 Ψ(x)

n = 8 ∫L8(x) · J0(x) dx =

=∫

(x8 − 64 x7 + 1568x6 − 18816 x5 + 117600 x4 − 376320 x3 + 564480 x2 − 322560 x + 40320) J0(x) dx =

= (7 x7 − 384 x6 + 7595 x5 − 66048 x4 + 238875 x3 − 224256 x2 + 40320x) J0(x)+

+(x8 − 64 x7 + 1519 x6 − 16512 x5 + 79625 x4 − 112128 x3 − 152145 x2 + 125952 x) J1(x) + 192465 Φ(x)∫L8(x) · J1(x) dx = (−x8 + 64 x7− 1520 x6 + 16576 x5− 81120 x4 + 127680 x3 + 84480 x2− 40320) J0(x)+

+(8x7 − 448 x6 + 9120 x5 − 82880 x4 + 324480 x3 − 383040 x2 − 168960 x) J1(x) + 60480 Φ(x)∫L8(x) · I0(x) dx = (−7 x7 + 384 x6 − 8085 x5 + 84480 x4 − 474075 x3 + 1428480 x2 + 40320x) I0(x)+

+(x8− 64 x7 + 1617 x6− 21120 x5 + 158025x4− 714240 x3 + 1986705x2− 3179520 x) I1(x) + 2027025Ψ(x)

133

∫L8(x) · I1(x) dx = (x8−64 x7 +1616 x6−21056 x5 +156384 x4−692160 x3 +1815552 x2 +40320) I0(x)+

+(−8 x7 + 448 x6 − 9696 x5 + 105280 x4 − 625536 x3 + 2076480x2 − 3631104 x) I1(x) + 2399040 Ψ(x)

n = 9∫L9(x) · J0(x) dx =

∫(−x9 + 81 x8 − 2592 x7 + 42336x6 − 381024 x5 + 1905120x4 − 5080320x3+

+6531840x2 − 3265920 x + 362880) J0(x) dx = (−8 x8 + 567 x7 − 15168 x6 + 191835 x5 − 1160064 x4+

+2837835x3 − 880128 x2 + 362880 x) J0(x) + (−x9 + 81 x8 − 2528 x7 + 38367 x6 − 290016 x5+

+945945x4 − 440064 x3 − 1981665 x2 − 1505664 x) J1(x) + 2344545 Φ(x)∫L9(x) · J1(x) dx =

= (x9 − 81 x8 + 2529 x7 − 38448 x6 + 292509 x5 − 982368 x4 + 692685 x3 + 1327104x2 − 362880) J0(x)+

+(−9 x8 + 648 x7 − 17703 x6 + 230688 x5 − 1462545x4 + 3929472x3 − 2078055 x2 − 2654208 x) J1(x)−

−1187865 Φ(x)∫L9(x) · I0(x) dx =

= (8 x8 − 567 x7 + 15936 x6 − 231525 x5 + 1906560x4 − 9188235 x3 + 25413120 x2 + 362880 x) I0(x)+

+(−x9+81x8−2656 x7+46305x6−476640 x5+3062745x4−12706560 x3+34096545x2−54092160 x) I1(x)+

+34459425Ψ(x)∫L9(x)·I1(x) dx = (−x9+81x8−2655 x7+46224x6−473949 x5+3014496x4−12189555 x3+30647808x2+362880) I0(x)+(9x8−648 x7+18585x6−277344 x5+2369745x4−12057984 x3+36568665x2−61295616 x) I1(x)+39834585Ψ(x)

n = 10∫L10(x) · J0(x) dx =

∫(x10 − 100 x9 + 4050 x8 − 86400 x7 + 1058400x6 − 7620480 x5 + 31752000x4−

−72576000 x3 + 81648000 x2 − 36288000 x + 3628800) J0(x) dx = (9 x9 − 800 x8 + 27783 x7 − 480000 x6+

+4319595x5−18961920 x4+30462075 x3+6543360 x2+3628800 x) J0(x)+(x10−100 x9+3969 x8−80000 x7+

+863919x6 − 4740480 x5 + 10154025 x4 + 3271680 x3 − 9738225 x2 − 49374720 x) J1(x) + 13367025 Φ(x)∫L10(x) · J1(x) dx = (−x10 + 100 x9 − 3970 x8 + 80100x7 − 867840 x6 + 4816980x5 − 10923840 x4+

+321300x3+5742720 x2−3628800) J0(x)+(10x9−900 x8+31760 x7−560700 x6+5207040 x5−24084900 x4+

+43695360x3 − 963900 x2 − 11485440 x) J1(x)− 35324100 Φ(x)∫L10(x) · I0(x) dx = (−9 x9 + 800 x8− 28917 x7 + 556800 x6− 6304095 x5 + 43845120 x4− 189817425 x3+

+495912960x2 + 3628800x) I0(x) + (x10 − 100 x9 + 4131 x8 − 92800 x7 + 1260819 x6 − 10961280 x5+

+63272475x4 − 247956480x3 + 651100275x2 − 1028113920 x) I1(x) + 654729075 Ψ(x)∫L10(x) · I1(x) dx = (x10 − 100 x9 + 4130 x8 − 92700 x7 + 1256640x6 − 10864980 x5 + 61911360 x4−

−235550700x3 + 576938880 x2 + 3628800) I0(x) + (−10 x9 + 900x8 − 33040 x7 + 648900x6 − 7539840 x5+

+54324900x4 − 247645440x3 + 706652100x2 − 1153877760 x) I1(x) + 742940100 Ψ(x)

n = 11∫L11(x) ·J0(x) dx =

∫(−x11 +121x10−6050 x9 +163350x8−2613600 x7 +25613280x6−153679680 x5+

134

+548856000x4 − 1097712000 x3 + 1097712000x2 − 439084800 x + 39916800) J0(x) dx =

= (−10 x10 + 1089 x9 − 47600 x8 + 1074843x7 − 13396800 x6 + 90446895 x5 − 293195520x4+

+289864575x3 + 150140160x2 + 39916800 x) J0(x)+

+(−x11 + 121 x10 − 5950 x9 + 153549 x8 − 2232800x7 + 18089379x6 − 73298880 x5+

+96621525x4 + 75070080 x3 + 228118275x2 − 739365120x) J1(x)− 188201475Φ(x)∫L11(x) · J1(x) dx = (x11 − 121 x10 + 5951 x9 − 153670 x8 + 2238687x7 − 18237120 x6 + 75325635 x5−

−111165120x4−32172525 x3−208391040x2−39916800) J0(x)+(−11 x10+1210 x9−53559 x8+1229360 x7−−15670809 x6+109422720x5−376628175x4+444660480x3+96517575x2+416782080x) J1(x)−535602375 Φ(x)∫

L11(x) · I0(x) dx = (10 x10 − 1089 x9 + 49200 x8 − 1212057x7 + 18043200 x6 − 170488395x5+

+1047755520x4−4203893925 x3+10577468160x2+39916800x) I0(x)+(−x11+121x10−6150 x9+173151x8−−3007200x7 + 34097679 x6 − 261938880x5 + 1401297975 x4 − 5288734080x3 + 13709393775x2−

−21594021120x) I1(x) + 13749310575 Ψ(x)∫L11(x) · I1(x) dx = (−x11 + 121 x10 − 6149 x9 + 173030 x8 − 3000987 x7 + 33918720 x6−

−258714225x5 + 1362905280x4 − 4978425375 x3 + 12000954240x2 + 39916800) I0(x)+

+(11x10 − 1210 x9 + 55341x8 − 1384240 x7 + 21006909 x6 − 203512320x5 + 1293571125x4−−5451621120x3 + 14935276125x2 − 24001908480x) I1(x) + 15374360925 Ψ(x)

n = 12∫L12(x) ·J0(x) dx =

∫(x12− 144 x11 +8712x10− 290400 x9 +5880600x8− 75271680 x7 +614718720x6−

−3161410560 x5+9879408000x4−17563392000x3+15807052800x2−5748019200 x+479001600) J0(x) dx =

= (11 x11−1440 x10+77319 x9−2208000x8+36293103 x7−345646080 x6+1803334995 x5−4350136320 x4+

+2588199075x3 − 325693440 x2 + 479001600x) J0(x)+

+(x12 − 144 x11 + 8591 x10 − 276000 x9 + 5184729x8 − 57607680 x7 + 360666999x6 − 1087534080 x5+

+862733025x4 − 162846720x3 + 8042455575 x2 − 5096632320 x) J1(x)− 7563453975Φ(x)∫L12(x) · J1(x) dx = (−x12 + 144x11− 8592 x10 + 276144x9− 5193240 x8 + 57874608x7− 365443200 x6+

+1135799280x5 − 1108771200 x4 + 526402800x3 − 6936883200x2 − 479001600) J0(x)+

+(12x11−1584 x10+85920x9−2485296 x8+41545920x7−405122256 x6+2192659200x5−5678996400 x4+

+4435084800x3 − 1579208400x2 + 13873766400x) J1(x)− 4168810800Φ(x)∫L12(x) · I0(x) dx = (−11 x11 + 1440 x10 − 79497 x9 + 2438400x8 − 46172511 x7 + 568673280x6−

−4689631485 x5 + 26293800960x4 − 99982696275x3 + 245477191680x2 + 479001600x) I0(x)+

+(x12 − 144 x11 + 8833 x10 − 304800 x9 + 6596073x8 − 94778880 x7 + 937926297x6 − 6573450240 x5+

+33327565425x4 − 122738595840x3 + 315755141625x2 − 496702402560x) I1(x) + 316234143225 Ψ(x)∫L12(x) · I1(x) dx = (x12 − 144 x11 + 8832 x10 − 304656 x9 + 6587160 x8 − 94465008 x7 + 930902400x6−

−6467685840 x5 + 32221065600x4 − 114578679600 x3 + 273575577600x2 + 479001600) I0(x)+

+(−12 x11 + 1584 x10 − 88320 x9 + 2741904 x8 − 52697280 x7 + 661255056x6 − 5585414400 x5+

+32338429200x4 − 128884262400x3 + 343736038800x2 − 547151155200x) I1(x) + 349484058000 Ψ(x)

135

a) Hermite Polynomials Hn(x) :

Weight function: w(x) = e−x2, interval −∞ < x < ∞.

Holds∫

H0(x)Zν(x) dx =∫

Zν(x) dx and∫

H1(x) Zν(x) dx =∫

2xZν(x) dx .

n = 2 ∫H2(x) · J0(x) dx =

∫(4 x2 − 2) J0(x) dx = −2xJ0(x) + 4x2 J1(x)− 6 Φ(x)∫

H2(x) · J1(x) dx = (−4 x2 + 2) J0(x) + 8xJ1(x)∫H2(x) · I0(x) dx = −2x I0(x) + 4x2 I1(x) + 2 Ψ(x)∫

H2(x) · I1(x) dx = (4x2 − 2) I0(x)− 8x I1(x)

n = 3 ∫H3(x) · J0(x) dx =

∫(8 x3 − 12 x) J0(x) dx = 16x2 J0(x) + (8x3 − 44x) J1(x)∫

H3(x) · J1(x) dx = −8x3 J0(x) + 24x2 J1(x)− 36 Φ(x)∫H3(x) · I0(x) dx = −16x2 I0(x) + (8x3 + 20x) I1(x)∫H3(x) · I1(x) dx = 8x3 I0(x)− 24x2 I1(x)− 12 Ψ(x)

n = 4∫H4(x)·J0(x) dx =

∫(16x4−48 x2+12) J0(x) dx = (48 x3+12 x) J0(x)+(16x4−192 x2) J1(x)+204Φ(x)

∫H4(x) · J1(x) dx = (−16 x4 + 176 x2 − 12) J0(x) + (64 x3 − 352 x) J1(x)∫

H4(x) · I0(x) dx = (−48 x3 + 12 x) I0(x) + (16 x4 + 96 x2) I1(x) + 108 Ψ(x)∫H4(x) · I1(x) dx = (16 x4 + 80 x2 + 12) I0(x) + (−64 x3 − 160 x) I1(x)

n = 5 ∫H5(x) · J0(x) dx =

∫(32x5 − 160 x3 + 120 x) J0(x) dx =

= (128 x4 − 1344 x2) J0(x) + (32 x5 − 672 x3 + 2808 x) J1(x)∫H5(x) · J1(x) dx = (−32 x5 + 640 x3) J0(x) + (160 x4 − 1920 x2) J1(x) + 2040 Φ(x)∫

H5(x) · I0(x) dx = (−128 x4 − 704 x2) I0(x) + (32 x5 + 352 x3 + 1528 x) I1(x)∫H5(x) · I1(x) dx = (32 x5 + 320 x3) I0(x) + (−160 x4 − 960 x2) I1(x)− 1080 Ψ(x)

n = 6 ∫H6(x) · J0(x) dx =

∫(64 x6 − 480 x4 + 720 x2 − 120) J0(x) dx =

= (320 x5 − 6240 x3 − 120 x) J0(x) + (64 x6 − 2080 x4 + 19440x2) J1(x)− 19560 Φ(x)∫H6(x) · J1(x) dx = (−64 x6 + 2016 x4 − 16848 x2 + 120) J0(x) + (384 x5 − 8064 x3 + 33696 x) J1(x)∫

H6(x) · I0(x) dx = (−320 x5 − 3360 x3 − 120 x) I0(x) + (64 x6 + 1120 x4 + 10800 x2) I1(x) + 10680 Ψ(x)

136

∫H6(x) · I1(x) dx = (64 x6 + 1056 x4 + 9168 x2 − 120) I0(x) + (−384 x5 − 4224 x3 − 18336 x) I1(x)

n = 7 ∫H7(x) · J0(x) dx =

∫(128 x7 − 1344 x5 + 3360 x3 − 1680 x) J0(x) dx =

= (768 x6 − 23808 x4 + 197184 x2) J0(x) + (128x7 − 5952 x5 + 98592 x3 − 396048 x) J1(x)∫H7(x)·J1(x) dx = (−128 x7+5824x5−90720 x3) J0(x)+(896x6−29120 x4+272160x2) J1(x)−273840 Φ(x)∫H7(x) ·I0(x) dx = (−768 x6−13056 x4−111168 x2) I0(x)+(128 x7 +3264 x5 +55584 x3 +220656 x) I1(x)∫H7(x)·I1(x) dx = (128 x7+3136x5+50400x3) I0(x)+(−896 x6−15680 x4−151200 x2) I1(x)−149520 Ψ(x)

n = 8 ∫H8(x) · J0(x) dx =

∫(256 x8 − 3584 x6 + 13440 x4 − 13440 x2 + 1680) J0(x) dx =

= (1792 x7−80640 x5+1249920x3+1680x) J0(x)+(256 x8−16128 x6+416640x4−3763200 x2) J1(x)+3764880Φ(x)∫H8(x) · J1(x) dx = (−256 x8 + 15872 x6 − 394368 x4 + 3168384 x2 − 1680) J0(x)+

+(2048x7 − 95232 x5 + 1577472 x3 − 6336768 x) J1(x)∫H8(x) · I0(x) dx = (−1792 x7 − 44800 x5 − 712320 x3 + 1680 x) I0(x)+

+(256x8 + 8960 x6 + 237440 x4 + 2123520x2) I1(x) + 2125200 Ψ(x)∫H8(x) · I1(x) dx = (256 x8 + 8704 x6 + 222336 x4 + 1765248x2 + 1680) I0(x)+

+(−2048 x7 − 52224 x5 − 889344 x3 − 3530496x) I1(x)

n = 9 ∫H9(x) · J0(x) dx =

∫(512x9 − 9216 x7 + 48384 x5 − 80640 x3 + 30240x) J0(x) dx =

= (4096 x8 − 251904 x6 + 6239232x4 − 50075136 x2) J0(x)+

+(512x9 − 41984 x7 + 1559808x5 − 25037568 x3 + 100180512x) J1(x)∫H9(x) · J1(x) dx = (−512 x9 + 41472x7 − 1499904 x5 + 22579200 x3) J0(x)+

+(4608x8 − 290304 x6 + 7499520x4 − 67737600 x2) J1(x) + 67767840 Φ(x)∫H9(x) · I0(x) dx = (−4096 x8 − 141312 x6 − 3585024x4 − 28518912 x2) I0(x)+

+(512x9 + 23552 x7 + 896256 x5 + 14259456 x3 + 57068064) I1(x)∫H9(x) · I1(x) dx = (512 x9 + 23040 x7 + 854784 x5 + 12741120 x3) I0(x)+

+(−4608 x8 − 161280 x6 − 4273920 x4 − 38223360 x) I1(x)− 38253600Ψ(x)

n = 10∫H10(x) · J0(x) dx =

∫(1024x10 − 23040 x8 + 161280 x6 − 403200 x4 + 302400 x2 − 30240) J0(x) dx =

= (9216 x9 − 741888 x7 + 26772480 x5 − 402796800 x3 − 30240 x) J0(x)+

+(1024x10 − 105984 x8 + 5354496x6 − 134265600x4 + 1208692800 x2) J1(x)− 1208723040Φ(x)

137

∫H10(x) ·J1(x) dx = (−1024 x10+104960x8−5199360 x6+125187840x4−1001805120 x2+30240)J0(x)+

+(10240x9 − 839680 x7 + 31196160 x5 − 500751360 x3 + 2003610240x) J1(x)∫H10(x) · I0(x) dx = (−9216 x9 − 419328 x7 − 15482880 x5 − 231033600x3 − 30240 x) I0(x)+

+(1024x10 + 59904 x8 + 3096576x6 + 77011200 x4 + 693403200x2) I1(x) + 693372960 Ψ(x)∫H10(x) · I1(x) dx = (1024 x10 + 58880x8 + 2987520 x6 + 71297280 x4 + 570680640x2 − 30240) I0(x)+

+(−10240 x9 − 471040 x7 − 17925120 x5 − 285189120 x3 − 1141361280 x) I1(x)

n = 11∫H11(x) ·J0(x) dx =

∫(2048x11−56320 x9 +506880 x7−1774080x5 +2217600 x3−665280 x) J0(x) dx =

= (20480 x10 − 2088960 x8 + 103311360x6 − 2486568960x4 + 19896986880x2) J0(x)+

+(2048x11 − 261120 x9 + 17218560 x7 − 621642240x5 + 9948493440 x3 − 39794639040x) J1(x)∫H11(x) · J1(x) dx = (−2048 x11 + 259072 x9 − 16828416 x7 + 590768640x5 − 8863747200x3) J0(x)+

+(22528x10 − 2331648 x8 + 117798912x6 − 2953843200x4 + 26591241600x2) J1(x)− 26591906880 Φ(x)∫H11(x) · I0(x) dx = (−20480 x10 − 1187840 x8 − 60057600 x6 − 1434286080 x4 − 11478723840x2) I0(x)+

+(2048x11 + 148480 x9 + 10009600 x7 + 358571520x5 + 5739361920x3 + 22956782400x) I1(x)∫H11(x) · I1(x) dx = (2048 x11 + 146432 x9 + 9732096 x7 + 338849280x5 + 5084956800x3) I0(x)+

−(−22528 x10 − 1317888x8 − 68124672 x6 − 1694246400 x4 − 15254870400x2) I1(x)− 15254205120 Ψ(x)

n = 12 ∫H12(x) · J0(x) dx =

=∫

(4096x12 − 135168 x10 + 1520640 x8 − 7096320 x6 + 13305600 x4 − 7983360 x2 + 665280) J0(x) dx =

= (45056 x11 − 5677056 x9 + 368299008x7 − 12925946880x5 + 193929120000x3 + 665280 x) J0(x)+

+(4096x12 − 630784 x10 + 52614144 x8 − 2585189376 x6 + 64643040000x4 − 581795343360x2) J1(x)+

+581796008640Φ(x)∫H12(x) · J1(x) dx =

= (−4096 x12+626688x10−51655680 x8+2486568960x6−59690960640x4+477535668480x2−665280) J0(x)+

+(49152x11 − 6266880 x9 + 413245440x7 − 14919413760x5 + 238763842560x3 − 955071336960x) J1(x)∫H12(x) · I0(x) dx =

+(−45056 x11 − 3244032 x9 − 215018496x7 − 7490165760 x5 − 112392403200 x3 + 665280 x) I0(x)+

+(4096x12 + 360448 x10 + 30716928x8 + 1498033152 x6 + 37464134400x4 + 337169226240x2) I1(x)+

+337169891520Ψ(x)∫H12(x) · I1(x) dx =

= (4096 x12+356352x10+30028800x8+1434286080x6+34436171520x4+275481388800x2+665280) I0(x)+

+(−49152 x11 − 3563520 x9 − 240230400x7 − 8605716480 x5 − 137744686080x3 − 550962777600x) I1(x)

138

n = 13 ∫H13(x) · J0(x) dx =

=∫

(8192x13−319488 x11+4392960 x9−26357760 x7+69189120 x5−69189120 x3+17297280 x) J0(x) dx =

= (98304 x12−14991360 x10+1234452480x8−59411865600x6+1426161530880x4−11409430625280x2) J0(x)+

+(8192x13 − 1499136 x11 + 154306560x9 − 9901977600 x7 + 356540382720x5 − 5704715312640x3+

+22818878547840x) J1(x)∫H13(x) · J1(x) dx =

= (−8192 x13+1490944 x11−151996416x9+9602131968 x7−336143808000x5+5042226309120 x3) J0(x)+

+(106496x12−16400384 x10+1367967744x8−67214923776x6+1680719040000x4−15126678927360x2) J1(x)+

+15126696224640Φ(x)∫H13(x) · I0(x) dx =

= (−98304 x12−8601600x10−723271680x8−34558894080x6−829690214400 x4−6637383336960x2) I0(x)+

+(8192x13 + 860160 x11 + 90408960 x9 + 5759815680x7 + 207422553600 x5 + 3318691668480x3+

+13274783971200x) I1(x)∫H13(x) · I1(x) dx =

= (8192 x13 + 851968 x11 + 88737792x9 + 5564123136 x7 + 194813498880x5 + 2922133294080x3) I0(x)+

+(−106496 x12−9371648x10−798640128x8−38948861952x6−974067494400 x4−8766399882240x2) I1(x)−

−8766417179520Ψ(x)

n = 14∫H14(x)·J0(x) dx =

∫(16384x14−745472 x12+12300288x10−92252160 x8+322882560x6−484323840x4+

+242161920x2 − 17297280) J0(x) dx = (212992 x13 − 38658048 x11 + 3937849344x9 − 248730273792x7+

+8707173995520x5−130609062904320x3−17297280 x) J0(x)+(16384 x14−3514368 x12+437538816 x10−

−35532896256x8+1741434799104x6−43536354301440x4+391827430874880x2) J1(x)−391827448172160Φ(x)∫H14(x) · J1(x) dx = (−16384 x14 + 3497984x12 − 432058368x10 + 34656921600x8 − 1663855119360x6+

+39933007188480x4−319464299669760x2+17297280)J0(x)+(229376x13−41975808 x11+4320583680x9−

−277255372800 x7 + 9983130716160x5 − 159732028753920x3 + 638928599339520x) J1(x)∫H14(x)·I0(x) dx = (−212992 x13−22257664 x11−2314211328 x9−145149548544x7−5081848611840x5−

−76226276206080x3 − 17297280 x) I0(x) + (16384 x14 + 2023424 x12 + 257134592x10 + 20735649792x8+

+1016369722368x6 + 25408758735360x4 + 228679070780160x2) I1(x) + 228679053482880Ψ(x)∫H14(x) · I1(x) dx = (16384 x14 + 2007040 x12 + 253145088x10 + 20159354880x8 + 967971916800x6+

+23230841679360x4+185846975596800x2−17297280) I0(x)+(−229376 x13−24084480 x11−2531450880 x9−

−161274839040 x7 − 5807831500800x5 − 92923366717440x3 − 371693951193600x) I1(x)

139

n = 15∫H15(x) · J0(x) dx =

∫(32768 x15 − 1720320 x13 + 33546240 x11 − 307507200x9 + 1383782400 x7−

−2905943040x5 + 2421619200x3 − 518918400x) J0(x) dx =

= (458752 x14 − 97714176 x12 + 12061163520x10 − 967353139200x8 + 46441253376000x6−

−1114601704796160x4 + 8916818481607680x2) J0(x)+

+(32768x15 − 8142848 x13 + 1206116352x11 − 120919142400x9 + 7740208896000x7−

−278650426199040x5 + 4458409240803840x3 − 17833637482133760x) J1(x)∫H15(x) ·J1(x) dx = (−32768 x15+8110080x13−1193287680 x11+118442987520x9−7463291996160x7+

+261218125808640x5 − 3918274308748800x3) J0(x) + (491520x14 − 105431040 x12 + 13126164480x10−

−1065986887680x8 + 52243043973120x6 − 1306090629043200x4 + 11754822926246400x2) J1(x)−

−11754823445164800Φ(x)∫H15(x)·I0(x) dx = (−458752 x14−56426496 x12−7106641920 x10−566071296000 x8−27179724902400x6−

−652301773885440x4 − 5218419034321920x2) I0(x) + (32768x15 + 4702208 x13 + 710664192x11+

+70758912000x9+4529954150400x7+163075443471360x5+2609209517160960 x3+10436837549725440 x) I1(x)∫H15(x) · I1(x) dx = (32768 x15 + 4669440 x13 + 701276160x11 + 69118832640x9 + 4355870238720x7+

+152452552412160x5 + 2286790707801600x3) I0(x) + (−491520 x14 − 60702720 x12 − 7714037760x10−

−622069493760 x8 − 30491091671040 x6 − 762262762060800x4 − 6860372123404800x2) I1(x)−

−6860371604486400Ψ(x)

140

XXXIII. Integrals of the type∫

x−1 Zν(x + α)Z1(x) dx∫J1(x)J0(x + α)

xdx =

x + α

α

(J0(x)J1(x + α)− J1(x)J0(x + α)

)∫

I1(x)I0(x + α)x

dx =x + α

α

(I0(x)I1(x + α)− I1(x)I0(x + α)

)∫

J1(x)J1(x + α)x

dx =

= −x + α

αJ0(x)J0(x + α)− x

α2J1(x)J0(x + α) +

x + α

α2J0(x)J1(x + α)− x + α

αJ1(x)J1(x + α)∫

I1(x)I1(x + α)x

dx =

=x + α

αI0(x)I0(x + α) +

x

α2I1(x)I0(x + α)− x + α

α2I0(x)I1(x + α)− x + α

αI1(x)I1(x + α)

141

XXXIV. Higher antiderivatives:

Φ(x) and Ψ(x) are the same as in I., page 3. See also [1], 11.2. .

J0(x) = d2

dx2

x2J0(x)− xJ1(x) + xΦ(x)

= d3

dx3

x3

2 J0(x)− x2

2 J1(x) + x2 − 12 Φ(x)

= d4

dx4

x4 − 2x2

6 J0(x)− x3 − 4x6 J1(x) + x3 − 3x

6 Φ(x)

= d5

dx5

x5 − 5x3

24 J0(x)− x4 − 7x2

24 J1(x) + x4 − 6x2 + 924 Φ(x)

= d6

dx6

x6 − 9x4 + 32x2

120 J0(x)− x5 − 11x3 + 64x120 J1(x) + x5 − 10x3 + 45x

120 Φ(x)

= d7

dx7

x7 − 14x5 + 117x3

720 J0(x)− x6 − 16x4 + 159x2

720 J1(x) + x6 − 15x4 + 135x2 − 225720 Φ(x)

I0(x) = d2

dx2

x2I0(x)− xI1(x) + xΨ(x)

= d3

dx3

x3

2 I0(x)− x2

2 I1(x) + x2 + 12 Ψ(x)

= d4

dx4

x4 + 2x2

6 I0(x)− x3 + 4x6 I1(x) + x3 + 3x

6 Ψ(x)

= d5

dx5

x5 + 5x3

24 J0(x)− x4 + 7x2

24 J1(x) + x4 + 6x2 + 924 Φ(x)

= d6

dx6

x6 + 9x4 + 32x2

120 I0(x)− x5 + 11x3 + 64x120 I1(x) + x5 + 10x3 + 45x

120 Ψ(x)

= d7

dx7

x7 + 14x5 + 117x3

720 I0(x)− x6 + 16x4 + 159x2

720 I1(x) + x6 + 15x4 + 135x2 + 225720 Ψ(x)

Let

J0(x) =dn

dxn

An(x) J0(x)−Bn(x) J1(x) + Cn(x) Φ(x)

(n− 1)!

,

then holdsA8 = x8 − 20 x6 + 291 x4 − 1152 x2

B8 = x7 − 22 x5 + 345 x3 − 2304 xC8 = x7 − 21 x5 + 315 x3 − 1575 x

A9 = x9 − 27 x7 + 599 x5 − 5541 x3

B9 = x8 − 29 x6 + 667 x4 − 7407 x2

C9 = x8 − 28 x6 + 630 x4 − 6300 x2 + 11025

A10 = x10 − 35 x8 + 1095 x6 − 17613 x4 + 73728 x2

B10 = x9 − 37 x7 + 1179 x5 − 20583 x3 + 147456 xC10 = x9 − 36 x7 + 1134 x5 − 18900 x3 + 99225 x

A11 = x11 − 44 x9 + 1842 x7 − 45180 x5 + 439605 x3

B11 = x10 − 46 x8 + 1944 x6 − 49770 x4 + 581535 x2

C11 = x10 − 45 x8 + 1890 x6 − 47250 x4 + 496125 x2 − 893025

A12 = x12 − 54 x10 + 2912 x8 − 100770 x6 + 1702215x4 − 7372800x2

B12 = x11 − 56 x9 + 3034 x7 − 107640 x5 + 1973205 x3 − 14745600 xC12 = x11 − 55 x9 + 2970 x7 − 103950 x5 + 1819125 x3 − 9823275 x

To get the functions for I0(x) one has to change all ’-’ in the fractions to ’+’ .The higher antiderivatives of J1(x) and I1(x) follow from the previous tables and the formulasJ1(x) = −J ′0(x) and I1(x) = I ′0(x).

142

IC. Miscellaneous:∫xn · J0(x) · J n−1

1 (x) dx =xn

nJ n

1 (x) , n = ±1, ±2, ±3, . . . ...∫xn · I0(x) · I n−1

1 (x) dx =xn

nI n1 (x) , n = ±1, ±2, ±3, . . . ...∫

Φ(x)x

dx = Λ1(x)− xJ0(x)− Φ(x)

143

bessel functions (tables of some indefinite integrals)

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