the evaluation of some definite integrals involving bessel ... · pressed in terms of complete...

The Evaluation of Some Definite IntegralsInvolving Bessel Functions Which Occur

in Hydrodynamics and Elasticity

By A. H. Van Tuyl

1. Introduction. Integrals of the form

(1.1) h(m.,v)= f e~xtJv(yt)JÁbt)tm dtJo

and

(1.2) h(m,v)= [ e~xtJ„-i(yt)Jv(bt)tm+idtJo

occur in various problems of potential theory, hydrodynamics, and elasticity (A.

Weinstein [1, 2], H. Bateman [3, p. 417], and A. G. Webster [4, p. 367-375]). In [1]-[4], v is an integer, x, y, and b are real, and m is a positive or negative integer, or

zero, such that (1.1) and (1.2) converge. When v is an integer, special cases of (1.1)

and (1.2) have often been expressed in the literature in terms of elliptic integrals,

the modulus of which is an elementary algebraic function of x, y, and b. The object

of the present paper is to obtain closed expressions for h(m, v) in terms of known

functions, assuming that v is complex when m ^ 0, and an integer when m < 0,

and to find recurrence relations involving I\(m, v) and /2(to, v). It is assumed

throughout that x, y, and b are complex. When m ^ 0, I\(m, v) is evaluated in

closed form in terms of associated Legendre functions of the second kind of the form

QTv-w(z); when to < 0 and v is an integer, it is expressed in terms of associated

Legendre functions of the form P~^!m(z), and Appell's generalized hypergeometric

function F i. By means of either the recurrence relations in Section 4, or certain

differentiation formulas, explicit expressions of the same form as in Sections 2 and 3

can also be obtained for 72(to, v). When v is an integer, these explicit expressions

for h(m, v) and h(m, v) can be obtained in terms of standard elliptic integrals by

means of well-known reduction formulas.

The more general integral

(1.3) I(ß, *;X)= f e'ctJß(at)JXbt)tx dt•>o

has been treated by G. Eason, B. Noble, and I. N. Sneddon [5] when p, v, and X are

integers such that (1.3) converges, and a, b, and c are real. A summary of [5] is given

by Y. L. Luke [6, p. 314]. The results of [5] can be used for the calculation of (1.3)

in terms of elliptic integrals for all integers ß, v, and X, X ^ 1, for which it converges.

However, explicit expressions are given in [5] only when (1.3) is of the form of

certain special cases of (1.1) and (1.2). The calculation of (1.3) for other values of

/x, v, and X can be carried out by use of recurrence relations obtained in [5]. The three

recurrence relations obtained in Section 4 of the present paper are seen not to be

special cases of the recurrence relations in [5].

Received August 22, 1963. Revised January 31, 1964.

421

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422 A. H. VAN TUYL

When v is an integer, and either m S: 0 or b = y, (1.1) and (1.2) can be ex-

pressed in terms of complete elliptic integrals of the first and second kinds. When

m < 0 and b = y, complete elliptic integrals of the third kind occur also. Examples

of the latter can be obtained from results of several authors (W. M. Hicks [7, p. 628],

G. M. Minchin [8, p. 354], A. Van Tuyl [9], M. R. Shura-Bura [10], M. A. Sadowskyand E. Stemberg [11], G. E. Pringle [12, p. 385 and 392], and H. E. Fettis [13]).In [7], [8], and [11], integrals of the form of (1.1) or (1.2) are not considered ex-

plicitly. A quantity known to be proportional to 72( — 1, 1) is evaluated in terms of

elliptic integrals in [7] and [8], and quantities proportional to 7i ( — 1, 1 ), 72 ( —T, 1 ),

and 72(—2, 1) are evaluated in [11]. The evaluation of 72( —1, 1) and 72(—2, 1)

by numerical integration has been discussed by G. P. Weeg [14].

For convergence of (1.1) and (1.2) at the lower limit, it is necessary to have

Re (2v + to + 1) > 0, and for convergence at the upper limit, all four quantities

Re (x ± ib ± iy) must be either positive or zero. In the latter case, we must also

have either to < 1 or to < 0, depending on the values of x, b, and y. When

Re (2v + m + 1) > 0 and Re (x ± ib ± iy) > 0, it is easily shown that (1.1) and

(1.2) are analytic functions of each of the variables x, y, and b, and that differen-

tiations of (1.1) and (1.2) with respect to x, y, and b can be carried out under the

integral sign.

Defining

(1.4) 4>(x, y) = y'-'him - 1, v),

(1.5) >p(x, y) = -yh(rn, v),

it is seen that <t>(x, y) and \p(x, y) satisfy the differential equations

(1.6) y2-ld^=f,dx dy

(1.7) y^f=-f.dy ox

Despite the fact that v may be complex, it is convenient, following A. Weinstein

[15], to call <t>(x, y) a generalized axially symmetric potential in 2v -4- 1 dimensions,

and \p(x, y) its associated stream function. The analogy with the case when v is an

integer has proven fruitful. Similarly, writing

(1.8) 4>(x, b) = b-yh(m, v),

(1.9) Hx, b) = -b'+lh(m,v + 1),

we have

(1.10) ^+id* = djP

dx dy

(1.11) ^+'^=_^.dy dx

In [l]-f4], integrals 7i(m, n) and 72(to, n) arise from the solution of special cases of

these equations.


SOME DEFINITE INTEGRALS INVOLVING BESSEL FUNCTIONS 423

We note that 72(»i, v) can be calculated in terms of 7i(m, v) by means of the

relations

(1.12) him, v) = y-'^- [yh(m, v)]dy

(1.13) = -b-'-lb'-'him^- 1)].

Thus, explicit expressions for 72(to, v), to ^ 0, and 72(to, n), m < 0, can be ob-

tained from the results of Sections 2 and 3 respectively. Alternatively, we can calcu-

late 72(to, v) in terms of 7i(to, v) by means of one of the recurrence relations in

Section 4.

2. Evaluation of 7i(to, v) for to ^ 0. From Watson [16, p. 389], equation (1), we

have

. / x (byyr(2v + to + l)him' v) = *x^^T(2v + 1)

f 2v + to + 1 2, + to + 2 p2\ . 2r

where F (a, b; c; z) is the hypergeometric function, and p2 = b2 + y2 — 2by cos <£.

Equation (2.1) holds for the same values of the parameters for which (1.1) and

(1.2) converge. From [6, p. 75], it follows that it is sufficient to assume that

| arg b | ^ 7r/2 and | arg y | g t/2. It is assumed in (2.1) that | arg (1 + p2/x2) \ <

■K, and that arg (sin <t>) = 0.

Setting b = c + n+ein [17, p. 109], equation (3), where n ^ 0 is an integer,

and where c and a — c — n are not negative integers, and letting e —» 0, we obtain

EV , \ r(c)r(c - a + n) , s_aF(a,c + n;c;z)= _ (I - z)

(2.2)

F ia, —n; a — c — n + 1;-)

when | arg (1 — z) \ < x. Hence, noting that the preceding conditions are satisfied

in each case, we have

„ (2v + to + 1 2v + to + 2*A-2-'-2-;" + 1;2

r9oN r(y + pr(j) n x-,-r-i,2(2-3) - r(„ + r + i)r(| ^7) (1 - z)

(2.4)Y(v + r + l)r(-i - r)

■F (v + r + |, -r; §; j-i-A to = 2r,

(i -^rr(*+ l)r(-§) „ ^-r-r-I/1

•FU + r + i'~r;I;r^>


424 A. H. VAN TUYL

r = 0,1, • • • , | arg (1 — z) | < r. From the preceding, using Legendre's duplica-

tion formula for T(z), we obtain

(-Dr2—lj2r(r + |)

, N l{ 'V) ~ T(byY^Y(v + h)

(2.5) , . . ,.^ . y M T(v + r + \ + ¿) i!\., , . .

' 5(_1) W-r(¥TÏ)-\2by) /(' + «> "}

and

(-l)'2r-'-1/2r(r + |)a;ii(,z/- -r x, j// =

(2.6)

/i(2r + 1, „) -

5 (_1) U-ñíTO-{¿by) I{,+1+ h V)>

r = 0, 1, • • • , where

sin2" </> d<t>(2.7) 7(to, ») = f

with

(2.8) /

(ß - cos4>)"<+''+1'2'

x + b2 + i/2

2by

We see that (2.7) is an analytic function of ß when the /3-plane is cut along the real

axis from 1 to — ». In order for (2.5) and (2.6) to be real when all parameters are

real and positive, we must then have | arg (ß — cos <£) | < x in (2.7). The cut

/3-plane is more extensive than the region in which (2.1) converges, hence, (2.5)

and (2.6) give an analytic continuation of (2.1).

Substituting cos0 = t in (2.7) and comparing with Hobson [18, p. 195], equation

(20), we obtain

\m 0M-l/2 Iap + |) , 2 Hs-mtfr.m(2.9) Km, v) = (-D"2'+1" xV^»'M0r - irml2QT-vAß)V(v + TO + %)

throughout the cut /3-plane, where Qa (z) is the associated Legendre function of the

second kind, and where [ arg (/3 — 1) | < x, | arg (ß + 1) | < x. We note that the

conditions arg (sin2 <¿>) = 0 and | arg (ß — cos 0) | < x which hold in (2.7) are

required in the preceding reference. Finally, from (2.5), (2.6), and (2.9), we have

<2J0) '■»■ '> " 2^m^ S 0) fâVî) (0 " - »"**•"«■<«and

t io„ _l i "\ 2 r(r + 1)War+ 1,0- - v(b y+w x

(2.11)

■emh)(Q,(ß2-ir^+1)K^^^^where ß satisfies the preceding conditions, and r = 0, 1, • • • .



The result

(by)" f" sin 0" d<f>(2.12) 7x(0, V)=^ML[

X Jo70 (z2 + b2 + y2 - 2by cos <t>Y+in

was obtained by A. Weinstein ([15], equations (9) and (12)), and the expression

(2.13) 7x(0, v) = ~^ Q,-i/2(/3)

was obtained by Watson ([16, p. 389], equation (2)).

When v is an integer n, n 3ï 0, we can express 7(to, v), and hence, Q7-w(ß), in

terms of complete elliptic integrals of the first and second kinds. Substituting

cos (¡> = 2sn u — 1 in (2.7) when v = n, where the modulus, k, of sn u is given by

(2.14) k2 = 2 áby

we obtain

1 + /3 x2 + (b + yy~'

„K in ïn jn-m+1/2, 2n+2m+l ¡ STl UCn U dU(2.15) 7(to, n) = 2n-m+ll2k2n+2m+l [

Jo dn2m+2nu

From (2.14), it follows that the cut in the ß-plane from 1 to — » corresponds to

cuts in the fc2-plane from 1 to =° and from 0 to — =° along the real axis. Concerning

the path of integration in (2.15), it is known that the residues of the integrand are

zero at all of its poles. Hence, it follows that (2.15) is independent of the path of

integration joining the points 0 and K. In particular, we can always take the latter

to be the straight line between 0 and K.

The right side of (2.15) can be evaluated in terms of elliptic integrals by means

of well-known reduction formulas (P. F. Byrd and M. D. Friedman [19, p. 191-

198]). Alternatively, we can compute Qn-i/2(/3) for to ^ 0 and n ^ 0 by use of the

recurrence relations for Q,(z) ([18, p. 290], equations (164) and (166)), starting

from the values of Q-m(ß), Qidß), Q-w(ß), and QU(ß). From (2.9) and (2.15),referring to [19], we have

(2.16) Q-i/»(0) - kK,

(2.17) Qm(ß) = I (K - E) - kK,

kk'E(2.18) Qlmiß) =

and

(2.19) QU(ß) = ¿-3 [2k'2K - (2 - k2)E],

where K and E are the complete elliptic integrals of the first and second kinds,

respectively, and k — 1 — k . Both k and k are single valued when the /c2-plane

is cut from 0 to — <x> and from 1 to ». Another method for the computation of

QZ-w(ß) in terms of elliptic integrals can be obtained from [7], Sections II and III.

Finally, we note that 7i(0, n) and 7i(l, n) satisfy simple recurrence relations.


426 A. H. VAN TUYL

From (2.11), we have

(2.20) h(l, n) = —7^ (ß2 - irmQn-uAß).ir(by)m

From (2.13), (2.22), and (2.16), we obtain the recurrence relations

(2.21) (n + \)h(fi, n + 1) - 2nßh(0, n) + (n - %)h(0, n - 1) = 0

and

(2.22) (n - i)7x(l, n + 1) - 2n/tfi(l, n) + (n + J)/i(l, n - 1) - 0.

Expressions for 7i(0, 0), 7i(0, 1), 7i(l, 0), and 7i(l, 1) in terms of elliptic integrals

ollow from (2.13), (2.20), and equations (2.16) through (2.19).

3. Evaluation of 7x(to, n) for to < 0. From [17, p. 105 and 107], equations (1),

(10), (18), and (37), we have the relation

(-z)"~c(l - z)c-a-hF(l - b,c - b;a + 1 - b; l/z)

rtlv _ r(l - c)T(a+ 1 - b) w(n h „ ^ r(c)r(i - c)T(a + 1 - b)[aÄ) - T(l - b)T(a + 1 - c) Ha'b>c'z) r(2 - c)T(c - b)T(a)

■e^-'V-'d - z)c-a-"F(l - a, 1 - b; 2 - c; z)

when Im z > 0. Substituting a = r — n -\- \, b = r — n, and c = 1 — n + e,

n ^ r 2: 1, e > 0, and letting e —> 0, we obtain

Fir - r + -,n — r + \;n + 1; z

T(n + l)T(r - n+J) ^r;|;l/.)

, s r(r)r(|)(3.2)

, iyT(n + l)r(r -n + è)r(n)

r(w - r + l)T(r)r(r + §)

•2~"(1 - 2)2r~"~1/2Wr - n + -, r - n; 1 - n; zj ,

Im z > 0. The second hypergeometric function on the right side of (3.2) is of the

form F(a, —to; —to — I; z), where m è 0,1 ^ 0, and is defined as in [15, p. 101],

equation (3). We see that it remains a polynomial throughout the limiting process

e —> 0. Noting that both sides of (3.2) are analytic in the z-plane cut along the real

axis from 1 to «>, it follows by analytic continuation that (3.2) is valid for

| arg (1 — z) | < x. Similarly, setting a = r — n — \,b = r — n, and c = 1 —

n + «, n ¡à r è 1, « > 0, and letting e —> 0, we have

F ( n - r + 1, w + r + - ; n + 1; 2 j

_ r(n + l)r(r-n-j) / 1. \-f(or(ï)V ' r'2'VV

(3.3)

+ (-Dr r(w + i)r(r - n - è)r(n)

r(w - r + i)r(r)r(r - i)

-n/-, \2r-n-3/2j-, / 1 , \•2 (1—2) 7*M ?• — n — -, r — n; 1 — w; z 1



when I arg (1 — z) \ < x. Finally, from (3.2), (3.3), and (2.1), using Legendre's

duplication formula for T(z) together with the functional equations, we find that

T/ o -\ 2r?i! ,, v-ii 0„_2r ^ (n — r + i)[h( -2r,n) = (by) < -2 x ¿_

x(2ra)! "' \ i^>(n - I - i)l(2i+ 1)1

(3.4) • gj J(n - r + i + 1, ») + 22'—2 (-~^JT ^^

•E^^'-;/' ](p}K(2r-n,r +i_o (n — r — !)'.( .)l \byj

i,n)

and

(3.5)

7,(1 - 2r,n) = ** <5„r /*"* g (b " '' + ¿)!x(2n) ! J \ f=ó (r - 1 - i) !(2i) !

2A' r/„ . , „• , , „ï o2—2(2w - 2r + 1)1(I)' J(n _ r + i + i, n) _ 2—2 (2"(-; + 1)! (26,)-,

y (r-l+t)! /2,2V' ,„ , . ■ ■• 2-, 7-rrr.-. . I r— 1 K(2r - ÎI - 1, r + i, n.

,=o (n. — r — i)\(2i + 1)! \6///

r = 1, 2, - • • , where

/"" '„2n . i ,sin 0 d4>

(a - COS 0)"1

and

1 (0 — cos 4>)'l~ sin " 4> d<j>(3.7) 7i(9,TO,n) = [

Jo

with

(3.8) a =

(a — cos </>)"

& + y

2by '

and with ß defined as in (2.8). It follows from (3.4) and (3.5) that q may be a

positive or negative integer, or zero, and that to ^ n in (3.6) and (3.7). We see

that (3.6) and (3.7) are analytic functions of a when the a-plane is cut along the

real axis from — 1 to 1, and that (3.7) is an analytic function of ß when the /3-plane

is cut from 1 to — ». In order for K(q,m,n) to be positive when a > 1 and ß > 1,

we choose | arg (ß — cos t¡>) | < x in (3.7).

Substituting cos <t> = t in (3.6) and comparing with [18, p. 195], equation (20),

we find that

271 + 1/2-p/ i i \Í \n -\- 2 ) A"-m+W)ri/ 2 , \ (7i-m)/2+l/4nra-n-l/2/ \

(,3.9) J(m,n) = -v-^-e (a — 1) v»-i/2 (a)(to)

when the a-plane is cut from 1 to — « along the real axis, with | arg (a — 1 ) [ < x

and | arg (a + 1) | < x. Using Whipple's relation in (3.9) [18, p. 245], equation

(92), we obtain

rotrw tí \ (2n)!x, 2 .(„_m)/2D_„ i a \(3.10) J(m>n) = ___(a _i) pn_m ^-—=j


428 A. H. VAN TTJYL

in the cut a-plane whenRe a > 0. With | arg (« — 1) | < x and | arg (a +1) | < x

as before, we see that Re (a/y/o? — 1) > 0 both for Re a < 0 and Re a > 0, and

that the imaginary axis is mapped twice onto the segment 0 ^ a/\/a? -1^1.

Writing

(3.11) J(m,n) =^-~J(m,n),

we can verify that

(3.12) j(m,n) = (a2 - I)1""'?;!, i-^==j , Re« > 0,

(3.13) = e^ml2(ai2 + \Yn^),2PT-m (y==) , *=±iai,*i> 0,

(3.14) = (-l)V - lY—^PT-mly^—^j , Re« < 0

when the a-plane is cut along the real axis from —1 to 1, where Pñ-m(x) is defined

in the usual way when 0 < x < 1 [18, p. 99]. From (3.10) and [18, p. 100], equation

(28), we find that

/ 2 t \{n—m)/2 n—m / . -\, / 2 -, \~i/2(3 15) Jim n) = (a ~ 1} Y (n-w + i)! _ (« - 1)KÖ.LO) JKm,n} {a + ^^—¡y {^ {n _ m _ i)[(n +i)Hl2i (a + ^/¿rTjy

in the cut plane when Re « > 0, and with n 2ï m and n ^ 0. Noting that both

sides of (3.15) are analytic when the a-plane is cut along the real axis from —1 to

1, it follows by analytic continuation that (3.15) holds throughout the cut a-plane.

It can be verified that (3.9) through (3.15) remain valid when to < 0, and that

(3.15) simplifies to 2~n when m = 0.

Similarly, substituting cos 4> = 2t — 1 in (3.7) and comparing with [17, p. 231],

equation (5), we have

„, , (2n)!T (ß + ir112Mq'm'n) = 2^W " («+D-

(3.16)

• F1(n + -,m,- - q,2n + 1;2' '2 " 'a+l'/3+l,

where Fi(a, b,b ,c; x, y) is the first of Appell's generalized hypergeometric functions

of two variables. From (3.4), (3.5), (3.11) and (3.16), we find that

7i(-2r,n) = -2-r(by)r^x

g (n-r + Ql (2x\jk(n-\- i)\(2i + l)l\by) J{n r + t + 1> n>

^r—Zn—312

(3.17) ' (2r- l)!n!

2"-3n-w(2n - 2r)l 1/2 ̂ r (r-l+i)! (tofV

,«„ in,„, ^V) ¿j (n-r-i)\(2i)l\by)

Fl(n + 5 r + i,n - 2r + -,2n + 1;(a + 1)^

2 2 \

«+ l'/3 + 1/*



7,(1 ~2r,n)

= 2-(%r £ ?(n;rt%, (^7 J(n - r + i + 1, n)i-o (r - 1 - z)!(2t)! \&?//

_ 23r-3n-"2(2n - 2r + 1)1 . _s/2 yT (r-l+i){ /2x2V'

(3.18) (2r-2)1*1 k J) X h (n - r - i)\ (2* + 1)! \by)

(ß + 1)2—3'2-, /,!.,,„ 0. , 3Fi(n + -,r + i,n-2r + ^,2n + l;

(a + l)r+i \ 2 ' '2

+ 1 ' ß + 1/ ' =

Finally, we consider the evaluation of K(q, to, n) in terms of elliptic integrals.

Let k be given by (2.8), and let

(3.19) Z2= 2 4by

a + 1 (6 + î/)2 "

We see that the cut in the a-plane from — 1 to 1 corresponds to a cut in the I -plane2

along the real axis from 1 to «j. Substituting cos <j> = 2sn u — 1 in (3.7), we have

22n-m+g+l,2z2ro K m2nu ^ ^2^ ^

(3.20) £(«,m,n)- ^ jJo (1 - Z2sn2w)m

When A; and I" lie in their cut planes, the path of integration in the w-plane corre-

sponding to 0 í= 4> á x is a simple curve which does not pass through any singu-

larities of the integrand. For values of k2 and t in the cut planes such that k2 ^ t,

k ?¿ 0, we see that (3.20) can be expressed in terms of complete elliptic integrals of

the first, second, and third kinds. When k = I, and hence, x = 0, only complete

elliptic integrals of the first and second kinds occur. Finally, when either b or y

vanishes, the integrand reduces to sin nu cos "u, and the upper limit becomes x/2.

Let

„K in in 7,__, \ fn \ C sn ucn udu(3'21) K(m,n) = I (1 _ lWu)m i

„K in in j/ooo\ ti \ r sn ucn udu(3-22) L(m,n) = l dntmu .

Then when q ^ 0, we have

^271-171+3+1/2, 2m-2g+l72m-25 t» / \ / ,2 ,2\t

(3-23) K{^n) = (P-V« £.U - *)t-¿(í>)'

and when g < 0, we find on expanding du2qu(\ — tsnu)~m in partial fractions that

r,2n-m+5+l/2, 27n-29+l727íi-2(¡

7í(g, to, n)(Z2 - fc2)m~q

(3.24) • {g (-ir+«+< (_;!\) £^£ l(z»

+ E( * .)^^7C(,»lírl \to — %) k2'


430 A. H. VAN TUYL

We see that (3.21) and (3.22) can be expressed in terms of integrals of the forms

»X nK

(3.25) Un= [ (l-l2snu)ndu, Vn = / (1 - l2snu)-n duJo ¿o

and

(3.26) un = I dn"udu, v» = / dn~2nudu,Jo Jo

respectively, where n Sï 0, and the latter can be evaluated by use of well-known

recurrence relations [19, equations (331.03), (336.03), (314.05) and (315.05)].

Substituting q = \ in (3.20) and letting k tend to zero with I fixed, we obtain

(3.27) J(m, n) = 22^l2m C ¿f^™^^ .Jo (1 — I sin2 u)m

Thus, an alternative method for the evaluation of J(m, n) is to express (3.27) in

terms of the integrals

-.W2 W2

(3.28) gn = / (1 — I2 sin u)n du, hn = / (1 — Z2sin2 uY" du,Jo Jo

and to evaluate the latter by means of recurrence relations obtained from those for

Un and Vn , respectively.

4. Recurrence Relations for h(m, v) and 72(to, v). Using the identities

(4.1) ^MZ) =V-J,(Z) -Jr+xU)

and

(4.2) £jM =J,-i(z)-VJ,(z),dz z

and assuming the conditions on x, y, and b mentioned earlier, we have

y f e-'JUbt) ¡JUyt) - (y~1)f-lWl tm+1 dt(4.3) Ja L u Vt J

= f e-*T+1JUbt) 1 JUyt) dt,Jo dt

b re-juyt)\(>--l)'rm - um(4.4) J° L m U

= f e-xttm+1J^(yt) | J^bt) dt,Jo at

and

b f e~xtJr(yt) \j,-Ábt) - "4+^1 tm+1dt

(4.5) J° L M J= f e-xttm+lJv(yt)d-Jv(bt)dt.

Jo dt

r+1 <ii



We note that any other limits of integration can be used in the preceding as well

as 0 and <». Integrating the right-hand sides of (4.3), (4.4) and (4.5) by parts,

and using (4.1) in the first two cases and (4.2) in the last, we obtain the relations

bli(m, v) = yh(m, v — 1) + (to + 1)7i(to, v — 1) — xl\(m + 1, v — 1)

0, Re (2v + m - 1) > 0

+(4.6)

(by)'2v + TO - 1 = 0,

(4.7)

and

(2v + to - 1)7!(to, v - 1)

= y [ e-xtJv(yt)J„-i(bt)tm+ï dtJo

22"-2[r(y)]2'

bh(m, v) — xh(m + 1, v — 1)

0, Re (2v + m - 1) > 0

(by)-1

22-2[r(v)] 2 '2v + m - 1 = 0,

(4.8)

(2v — to — 1)7i(to, v) — yh(m, v) + xlx(m + 1, v)

= b [ e-xtJ„(yt)J„-i(bt)tm+1 dt, Re(2,/ + to + 1) > 0,Jo

respectively. Multiplying (4.7) by b and (4.8) by y and subtracting, we obtain

(if — b2)h(m, v) = (2v — to - l)yh(m, v) 4- xyh(m + 1, v)

-(2v + to - l)bh(m, v - 1) + zWiXto + 1, v - 1)

(4.9) 0, Re (2v + to - 1) > 0

2v + to - 1 = 0..22"-2[r(iv)]2'

Finally, from (4.6) and (4.9), we find the recurrence relation

(2v — to + l)byh(m, v + 1) = 2v(y + b2)h(m, v)

— bxyli(m + 1, v + 1) — (2v + to — \)byl\(m, v — 1)

(4.10)

+ bxyl\(m -\- 1, v — 1)

0, Re (2)v + to - 1) > 0

(&2/r+122"[r(,.+ i)]2'

2¡> + to - 1 = 0

for 7i(to, v) alone.

It is easily verified that when 7j.(to, n) is known for 2n + to = 0, 1, and 2, to ^

— 1, and when 7j.(0, n) is known for n ^ 0, we can obtain I\(m, n) for all other

values of to < 0 and n ^ 0 for which it converges by means of (4.10). We can then

obtain 72(to, n) from (4.9). Also, when 7j.(to, 0) and 7i(to, 1) are known for m^0,

and when 7i(0, n) is known for n ^ 0, we can calculate 7i(to, n) for all other values

of to > 0 and n > 1 by solving (4.10) for 7i(to + 1, n + 1). As before, we can

then obtain 72(to, n) from (4.9).

U. S. Naval Ordnance Laboratory

White Oak, Silver Spring, Maryland


432 A. H. VAN TTJYL

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the evaluation of some definite integrals involving bessel ... · pressed in terms of complete...

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