handbook of numerical calculations in engineering
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Abou t
th e
Author
Jan J. Turna, Ph.D., is
Profcssor
of
Enginccring
at
Arizona
State
university. He has extensive expcriencc as an enginccring consultan
and has solved many problems on frame, piale , and space structures.
some of w hi ch h av e b ec om e landmarks of t h e Ameri can Southwes t .
Dr. Tuma is ihc
author of numerous
research papcrs
an d
scveral
volumes in McGraw-HiH's Schatim Outline Series.
Among
his
many
other
WOfks are the
Engineering
Mathematics Handbook,
Technolog\
Mathematics Handbook, Handbook of
Physical Calculations,
and the
Handbook
of Struclural and Mechanical Matrices, all publ is he d by
McGraw-Hi l l .
P
Con t e n t s
Preface ix
1 .
Numerical Calculat ions
1
2 . Numerical Constants 9
3 . Numerical Differences 45
4.
Numerical Integráis 67
5. Series and Products
of
Constants 85
6.
Algebraic
and Transcendental Equations «119
7. Matrix Equations
143
8. Eigenvalue Equations 157
9 .
Series of Functions
169
10.
Special Functions
«195
11. Orthogonal Polynomials 215
12. Least-squaresApproximations 237
13.
Fourier Approximations
249
—
14.
Laplace Transforms
267
15.
Ordinary Differential Equations 283
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The subjcct
material
is divided into
sixteen chapters
covering the
following:
(a)
Evaluation of numerical constants
(b)
Approximations of elementary and advanced functions
(c) Numericaldiflerentiationand integration
(d)
Solutions of algebraic and transcendental equations
(e) Solutions of systcms of equations
(/ ) Applications of Fouricrseries and Laplacc transforms
(g) Solutions ofordinaryand partial difTcrcntial equations
Finally, ten-digit tablcs of numerical valúesof the most important functions,
which cannot be displayed by hand-hcld calculators, are asscmblcd in the
Appcndixcs.
The
form
of presentation has thesame special fcatures as the mathemati-
cal
handbook mentioned
beforc.
(1) Each page is a table,
dcsignated bya tille and
section
number.
(2)
Lefl and righl
pages presen related or similar material, and important
modcls are placed in blocksallowing a rapid location of the dcsired
in fo rmation .
(3)
Numerical
examples are
located
below the respective
formulas,
providing a
direct i llus tra tion of the application on the same page or on the
oppositc page.
Although every cflbrt wasmadeto avoiderrors; it would be presumptuous
to assume that nonc had cscapcd dctcction in a work ot this scopc. The
author carncstly solicits commcnts and recommendations for improvements
an d futurc addi l ions .
In
closing,
gratitudc is expressed to Mrs. Ailccnc Sparling
who
typed
the
final draft of the manuscript and to my wife Hana. for her paticnce,
understanding and cncouragcmcnt during tlic preparationof this book.
Tempe, Arizona
Ja n
J.
Turna
Proface
* — -1
1
NUMERICAL
CALCULATIONS
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— 1
2 1 .01 CLASSIFICATION OF METHODS
Numerica l Calcula t ions
(a)
Methods
Of calculation fall into two major
cateogries,
designated as
analytical
methods and
numerical methods.
Analytical methods use algebraicand transcendental functions in the solution
of
problems, whereas the numerical methodsusearithmelic operations only.
(b)
Numerical
methods whichform the majorpartof this bookare classifíed as:
(1) Approximations
of
constants
(6) Operations with seríesand producís
(2) Approximations of functions
(7) Solutions of algebraic equations
(3) Approximations of derivatives
(8) Solutions of transcendental equations
(4) Approximations of integráis
(9) Solutions
of
systems
of
equations
(5) Summations of series and producís
(10) Solutions of diflerential equations
Two frequently usedabbreviations areCA = computer application andTF = Ulescopic form.
1.02 APPROXIMATIONS OF CONSTANTS
(a)Three types Of constants oceur in the solution of enginccring and applicd science problems.
They are the fundamental physical constants, the basic numericalconstants, and the derived
numer ica l cons tan ts .
(b) Fundamental physicalconstants are produets of natural laws. andas such theirvalúes canonly
beobtained by measurcments. A completelist of thcseconstantsis given in Appendix B, and in
the back endpapcrs.
(c) BasiC numerical constants are
special
numbers given by their definitions and frequently
oceurring in numerical calcuations. They can be evaluated to a dcsired degrecof aecuracy as
shown in sections indicated below. They are:
(1)
Slirling's numbers
SrY\
9\»
(2.11)
(5) Natural logarithm of 2 (2.20)
(2) Archimcdes constantn (2.17)
(6) Fibonacci numbers F, (2.21)
(3) Eulcr'sconstant ¿(2.18)
(7) Bcrnoulli numbers
B,,B,
(2.22)
(4) Euler'sconstant y (2.19)
(8) Eulcr numbers
E,, E,
(2.24)
(d)
Derived
numerical
constants are
special
numbers
obtained
as
particular valúes
of
certain
functions and
again
oceurring frequently in
numerical calculations.
They canalsobe calculated to
a
dcsired degrec
of
aecuracy
asshown in sections indicated
below.
They are:
(1)
Gamma functions ofintegerargument (factorials) T(n+ I) = n
(2.09), (2.28)
(2) Double factorials (2n)ü,(2n + I)ü (2.30)
(3) Binomial
coefficients
í j (2.10)
(4) Riemann zeta functions ofinteger argument Z(n), Z(n)
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4 1.05 APPROXIMATIONS OF INTEGRALS
Numerical Calculat ions
(a)
Algebraic polynomials are the only
functions
which can be intcgrated in given limits in a direct
and closed form.Integráisof allother functions must beevaluatedby somcapproximations.They
ar e
c l a ss i f ied a s :
(1)
Closed-form
integráis evaluated by functional approximations (1.03)
(2) Numerical
quadratures in
tcrms
of
diflcrcncc polynomials (4.05)
to
(4.09)
(3) Numericalquadratures in tcrmsofasymptotic series (4.10)
(4)
Numerical
quadraturesin tcrmsoforthogonalpolynomials (4.11) to (4.16)
(5) Integration by powcrseries expansión (11.01)
(6) Integration by trigonometricexpansión (13.14)
(7) Goursat's formulas (4.17)
(8) Filon's formulas (4.18)
(9) Multiplc-integral reduction formulas (14.03)
(b) Closed-form
integráis
expressed in terms ofelementan' transcendental functions orin tcrms of
combinations of algebraic and elcmenlary transcendental functions are
listed
in standard integral
tablcs
(Refs. 1.01, 1.20, 4.02, 4.03, 4.06) and as
such must
be evaluatcd by the approximations
given in (1.03).
(c) Numerical
quadrature formulas (4.05) to (4.09)
use either
valúes ofthe
integrand
at equidistant
spacing or
use
the
diíferencc
interpolation polynomial as the substitute function. Their powcr is
their
simplicily. The trapezoidal rule and Simpson's rule are the most commonly uscd
methods
in
t h i s c l as s .
(d)
Euler-MacLaurin
formula
adds correction
tcrms
ofasymptotic
series
tothe
trapezoidal
ruleand
provides the numerical analysis with one of the most important
rclationships,
which is
uscd
invcrselyin the summation of series (4.10).
(e) Gauss' Integration formulas, known
as
Gauss-Chebyshev,
Gauss-Legendre, Radau and
Lobatto formulas, use orthogonal polynomials as the basis for the cvaluation (4.11) to (4.16).
They gained receñí popularity
in
connection with the
finitc-element
methods.
(f) Power series expansión ofthe
integrand allows
the
cvaluation
of integráis of functions
which
cannot be evaluatcd in terms of elementary functions. The advantage of this method is i ts
application inthe cvaluation ofmúltiple integráis anditssimplicity in
handling
in general.
(g)
Trigonometric
series
expansión
ofthe
integrand
oífers the
same
advantages
asthe
powcr series
method. In addition the trigonometric expansión of the integrand gives a
symbolic
expression
for
the integral cvaluation.
(h)
Goursat's
formulas
allow the
evaluation
of
product
functions by a finitc series and are
particularly uscful in producís ofalgebraic and elementan- transcendental functions.
(I)
Filon's formulas
ofler
the approximatc evaluation
o(
definite integráis
oíf(x)costx
and
f(x)
sin
tx,
where / isa constant.
(j) Multiple-integral reductionformulas reducethe múltiple integral to a singleintegral by mcans of
the
convolution theorem (14.03).
«• i Mi
Numerical Calculations
(a)
Series
and
producís are fmite or infinite, and they are classified as:
(1) Series and produets ofconstants(Chap. 5)
(2) Series
and produets of
functions
(Chap. 9)
mm m
«•*)
mm
1.06 SERIES AND PRODUCTS 5
(b) Sums
ofseries
and produets
ofconstants are calculated by:
(1) Fundamental theorem ofsum calculus
(5.03)
(2)
Diíferencc
series formula
(5.04)
(3) Powcrseriesformula (5.04)
(4) Euler-MacLaurin
formula (5.05)
(5) General and special transformations (5.06),
(5.07)
(c)
Sums
of series of
special
constants
are
available
for:
(1) Arithmeticand geometricseries(5.08)
(2) Arithmogeomctric series(5.09),(5.10)
(3) Seriesof
powcrs
of integers (5.11). (5.12)
(4) Harmonic series of integers (5.13). (5.14)
(5) Series of
powcrs
of reciprocal integers (5.15),
(5.16)
(6) Seriesoffactorial polynomials(5.17),(5.18)
(7) Series of binomial coefficients (2.15). (2.16), (5.17), (5.18)
(8)
Series
ofpowcrs of numbers
(5.19).
(5.20)
(9) Harmonicseriesofdecimalnumbers
(5.21)
(10) Seriesofpowcrsof reciprocalnumbers (5.22)
(d)
Sums
ofseries ofproduets ofspecialconstants are available for:
(1) Series of produets of numbers (5.25). (5.26)
(2)
Series
of produets of fractions (5.27). (5.28)
(e)
Series Of functions are the power series and the transcendental
series.
The most important
series
in this group are:
(1) Algebraic powcr series(Chaps.9,
10)
(2) Finite-diflcrence powcrseries (Chap. 9)
(3) Fourier series (Chap. 13)
(f) Algébrale power
series
areeither the result ofcvaluation ofa function in
tcrms
of
(1) Taylor's seriesexpansión (9.03)
(2) MacLaurin's series expansión (9.03)
or
they
are
the
solutions ofdiflerential equations dcsignated by
special
ñames such as
Bcssel
functions
(10.07) to (10.17), Legendre functions (11.10),
Chcbyshev
functions (11.14),
Laguerre
functions (11.16), Hcrmite functions(11.17),and many more.
(g)
Finite-difference power
series
(9.03) may also be
used
for the cvaluation
of
functions, but
their
applications
yield only a
particular valué
ofthe function at a point.
(h) Fourier
series
are used for the same
purpose as
the algebraic
powcr
series. They
can
be used
in
the continuous range and
also
over equally spaced valúes in that range. They are
particularly
useful
in the
solution
of partial diflerential equations (13.01), (13.17).
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6
1.07 OPERATIONS WITH SERIES
AND PRODUCTS
i l
Numerica l Calcula tions
(a)
Nesting
Of power series for the purpose of summation isthe most commonly uscd operation, in
which
the
given series
is
rcplaccd
by a
nested product
(Rcf.
1.20,
p.
102)
as
É (±i)V
=
4 . onl y
special
cases can
be solved in closed
form.
In
general, the
higher-order
algebraic equations c an b e
solved
only by approximations.
(b) General Closed-form
SOlutionS
are
available for the
following algebraic
equations:
1.08
SOLUTIONS OF EQUATIONS
7
(1) Quadratic equations (6.02a)
(2)
Binomial equations
(6.02¿)
(3)
Trinomial
equations (6.02/)
(4) Cubicequations
(6.03a)
(5) Quarticequations
(6.04a)
(C)
Special Closed-form
SOlutionS are
available
for
the
following
algebraic equations:
(1) Symmetrical equations of
fifth
degree
(6.06a*)
(2)
Antisymmetrical
equations
offifth degrec (6.060
(d)
General
methods
used in the approximatc solution of algébrale and transcendental equations
are
based ontheconcept of
iteration
and
intcrpolation.
(6) Iteration methods are
applicable
in solutions
of
equations
of all
types, and the
most
important
methods
i n t hi s
group
are:
(1) Bisection method
(6.08)
(2) Sccantmethod (6.09)
(3) Tangcntmethod
(6.10)
(4)
General
iteration method
(6.11),
(6.12)
(f)
Polynomial
methods
of practical
¡mporunce
uscd
in the
scarch oí
real
and
complex roots
of
algebraic equations of higher degrec arer
;l) Newton-Raphson's method (6.16)
(2)
Bairstow's
method
(6.17)
Methods dcveloped in the era
of
hand calculations, such as Horner's method, GraeÜVs method,
Bcrnoulli's
method,
and Lagucrrc's method, are oriimitcd valué in
computer apphcations
and are
not covered i n t hi s
book.
1.09 SOLUTIONS 0FSYSTEMS 0FEQUATIONS
(a) Systems
Of linear algebraic equations
are classified as nonhomogencous
and
homogeneous.
Methods
of
solution
of
nonhomogencous equations fall
intotwo
categones:
(1) Direct
methods, producing an exact
solution
by
using
a finite
number
of
arithmctic
operations
(2) Iterative
methods, producing an
approximatc
solution of
dcsired aecuracy by y.cld.ng a
sequence
oí
solutions
which converges
to
the
exact solution
as
the
number
of
itcrations
tends to infinity
(b)
Direct methods
introduced
in this book are:
(1)
Cramer's rule
method
(7.03)
(2)
Matrix inversión
method
(7.04).
(7.05)
(3) Gauss
elimination
method
(7.06)
(4) Succcssive transformation
method (7.07)
(C) Iterativo
methods introduced
in
this book are:
(1)
Gauss-Seidcl
iteration method (7.13)
(5) Cholesky method
(7.09)
(6) Square-root
method
(7.11)
(7) Inversión
by partitioning
(7.12)
(2) Carryover method (7.14)
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1
2
P
O
N
B
C
N
m
c
C
a
s
(
a
P
o
o
a
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a
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4
a
)
(
4
a
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+
a
)
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d
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<
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2
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b
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+
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=
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h
b
a
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=
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K
H
=
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s
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g
s
n
H
>
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r
D
+
X
+
K
+
(
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T
1
0
c
o
g
é
i
n
d
i
n
(
2
0
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r
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y
a
c
a
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h
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o
g
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n
e
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s
2
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W
P
0
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H
•—1
~w¡
14 2.07 CONTINUED FRACTIONS, BASIC CONCEPTS
(a) Continuadfraction is of the form
Numerical
Constants
where the right-side expression is a symbolic reprcsentation.
(b) NumberOfterms ay,bx,a2,b2,... ,aK,bm
defines the continued fraction as terminating (n < °°)
or inf in it e (n = °°) .
(c) Simple COntlnued fraction has all partial numerators cqual to I.
I
f, =
4 ,+
b, +
1
¿,
+
* ,+
0
r¿
j l i
L
'b.+ 'b,
+
'¿,+
*
J
(d) Successive convergents for ak > 0, bk > 0 are defined as
/=o = 7 7 =
*o
^. = 77 =
*o
+ 7* = —:
Qo
C¿i
b, b.
_ _
/j
_ . a, MMj 4- a.,) + a,¿ ,
*i +
*2
F_ P* _ a*P*-* +M1-.
* a* «iai-2 + *iGi-,
an d
laJ
tai-, c J U J
where P_, = 1,
P„
=
b0, Q...
= 0,
&> =
1.
(e) Convergent continued fraction. If
/>
lim Fm = lim -=• = F (- » < F < »)
« - ♦ ■ « »-«o12.»
exists, the fraction is convergent. If ak = 1 and
bk
are integers, the continued fraction is always
convergent. lfak,bk are positiveintegers or fractions and
ak :£ bk
bk >
0
then the continued fraction is convergent. If
T« a\ ai fla 1
L A,
4-
b¡+ bs + J
and - |«¿ |-M-s-l>í|—--(* =-1, -2,3,—.)
then this conünucd fraction converges and its
absolute valué does
not cxcced unity.
rS-j
— i ~1 •— 1
208~
CONTINUED FRACTIONS,
is
NUMERICAL
PROCEDURES
umerical
Constants
(a)
Conversión Of
COntlnued fraction to a simple
fraction is obtained by
performing all
the
operations indicated
by
the continued fraction.
(b) Example
i
ii
i 1 43 931
5 +
6 +
*
(C) Any pOSltive rational number may be converted to a
fini.e
continued
fraction by
a
reversed
process.
(d) Example
i + 5
(e)
Any
positive
Irrationa.
number
may be
converted
to
an infinite
continued
fraction
by
the
same
process.
or
it may be approximated by the convergent
Fk
(2.07rf).
(1)
Fk
of odd
order
is
greater than
F.
but
decreasing.
(2) Fk
of even
order is iess than Fm but
increasing.
*
X - 1
l- l
k
*
-1
0
1
2
3
*- 2
«4
0
«i
fl2
«i
*4
1
0
1
P,
a.
P,
A -,
a» - i
/V-i
d i - i
\—'—
From this table, P_, =
1,
G-. = , P0 = ¿o, Go = Uand rccurrently,
gktc*
la-iuia-
la-i^^i
(g)
Example.
If
x
=3.141
59,
the corresponding truncated
continued fraction is
1
3 +
7 +
1
15 4-
1
1 4-
25
+
A„ = 3
1-H
1/0.14159=
7.062
65
1/0.062
65= 15.962
80
1/0.962 80=
1.038
64
1/0.03864 = 25.878 79
1/0.878 79= 1.137 93
1/0.137 93= 7.25008
and with
a,
=a3 =
a3
==
«i
= 1.
the
convergents
calculated
by (/) are
F4 = P4/& = 9208/2831 = 3.141 59
F, =
/VQi
= 9563/3044 = 3.141 59
* . -
7
b, = 15
¿3 = 1
64 = 25
A4 = 1
*» -
7
1
F, = PJQ.X = 22/7 = 3.142 86
F,= /VGi = 333/113 = 3.141
51
Fs = PJQ¿ = 355/113 = 3.HT59
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16
2.09
FACTORIALS
(a) Factorialof positive integer
n
¡sdefincd as
n = »(« - I)(h - 2)
3-2-
1
=
n(n -
1)
= „(,, - \)(n
-
2)
Numerica l Constants
andbydcfinition,
0
=
1.
Numerical valúes ofn
for n =
1,2,3,..., 100 are tabulatcd in
(A.03).
(b) Factorial ofproper
fraction (/(O
< U 0)
where N —n = u and n i sa posit ive integer nearest to
.Y.
(d) Example. For
N =
3.785.
n =
3,
u =
0.785.
A - (3.785)(2.785)(1.785)(0.785) = 17.410416
where from (A.04), u = 0.927 488.
(e) Factorialof N> 4 can be approximatcd by Stirling's expansión
N\ =
•«•©'
v/2AJr +
e
\ e \ < .
x I0 10
where n = Archimcdes constant (2.17), e = Fulcr 's const an t (2. 18) , and in te rms of Bernoul li
numbers/},,,/}4./}ri,5li (2.22),
A»2^
B2,
B,.\'-'
/}4.V '
B..N
' BnN~7
+ —
+ —
,r,2r(2r
- 1)ArJ'-' 1-2 3-1 5 6
_N^__N^ A^ _ A^
12 360
1260
1680
1 / S / 2S i 3SU\
I,,
nested
form, A=— (l -~(l -y( l - j)l)
(f)
Example.
For
A'
=
7.3.
with5 = (7.3)-' ,
A =
12(7.3)
V 30V 7 V 1
7. 3
By (e). A = / ' *' (
— Vl4.6,T
= 9281.39
with
S
= A'
By
(c),
A = (7.3)(6.3)(5.3)(4.3)(3.3)(2.3)(1.3)(0.3> = 9281.39
where rrom (A.04), (0.3) = 0.897 471.
(g) Factorial ofN > 100can be approximatcd by Stirlinv;s formula
'A\A'-,
A
=(-)
\¡2Ññ +e \e\ <
5x
lO '
This formula applics part icularly well in computing ratios
of
two factorials.
Numerical Constants
(a)
Notation.
x- signed number
k,m,n
= positive integers
(b) Binomial coefficient in xand k
is
by definition
(x(x- l)(x-2) (*-* + 1) 1}
k\
1 (* = 0)
(k =
I)
2.10 BINOMIAL
COEFFICIENTS i?
an d
for k > 0,
(x +
1
V
k
+
(I)
k + 1
x +
1
v* -
1/
U + 1
C+l)
n-k
\k-
¡j
)-o.*(*;.)
- /t 4 - 1
(c) Example
/-6.4X (-6.4)(-7.4)(-8.4>
=
^^
l 3 / (3)(2)(I)
(d) Binomial
coefficient in
n
and
k
is by
definition
(« - k)\k\
n(n -
Din
- 2)
(n
-A-+ 1)
/.-
id
for 0 <
k:
S n,
CK-,>
* -
n -
1
(«> *)
(l»l > *)
(k
= 1 or
= n
-
1)
(A
= n or A- = 0)
(k < 0)
¿4-1/
A+ lU/ ( l) '
. / \ -
, i io
9
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1
2
1
B
N
M
A
F
U
O
B
C
C
N
m
c
C
a
(
a
F
o
a
p
y
m
a
s
X
n
e
v
a
e
l
2
3
>
e
b
n
o
A
)
=
*
n
=
=
í
»
A
2
=
X
2
=
x
-
=
<
>
+
*
A
-
<
»
=
^
=
x
-
)
x
-
=
*
+
x
+
x
A
T
=
*
¿
=
x
-
)
x
-
2
x
-
p
-
=
»
+
<
+
+
V
w
e
&
s
e
S
r
n
m
o
h
r
k
n
A
0
(
b
F
e
o
o
m
a
I
5
a
k
m
a
e
w
h
o
m
2
*
0
¿
P
C
+
=
y
_
m
X
m
(
O
=
0
1
2
)
w
e
o
k
m
S
0
=
I
a
o
k
0
o
k
m
+
1
¡
*
(
C
E
m
e
I
y
2
=
-
1
=
+
h
n
o
S
=
í
^
=
^
2
2
=
+
y
=
y
-
2
-=
-
3
y
=
I
(
d
P
w
X
i
n
e
m
o
a
o
a
p
y
m
a
s
A
A
2
,
A
(
e
x
=
y
W
+
A
x
=
&
X
+
W
»
A
x
=
A
5
+
»
X
™
4
+
r
X
w
e
y
s
e
S
r
n
m
o
h
n
A
0
(
e
S
e
o
o
m
a
I
S
L
a
e
w
h
o
o
a
*
0
g
m
U
+
)
(
m
=
1
2
)
w
e
o
k
m
§
=
a
o
k
k
m
^
=
i
(
E
m
e
I
9
4
1
a
*
=
1
h
n
o
9
>
&
*
=
*
+
&
>
1
9
*
&
+
2
&
=
3
^
^
+
^
N
m
c
C
a
2
1
B
N
M
A
F
U
O
O
O
1
(
a
S
m
O
b
n
m
a
u
o
I
Q
n
*
a
e
b
n
m
a
f
u
o
n
a
e
v
a
b
e
x
n
a
a
e
v
n
e
p
h
(
K
=
+
=
[
+
]
\
¿
\
?
P
?
w
e
a
s
o
o
k
q
(
b
D
v
v
o
*
a
e
M
-
1
*
w
í
^
=
O
*
^
A
-
e
A
r
*
*
p
%
\
2
k
a
n
a
4
C
=
P
I
L
W
(
n
(
n
w
e
Z
=
d
d
1
2
3
(
c
n
n
o
n
e
á
s
o
a
e
a
n
a
4
3
¿
1
(
*
4
C
x
1
C
x
2
4
C
f
o
n
$
>
w
e
=
<
&
n
2
3
a
C
C
C
e
h
a
s
o
n
e
a
o
(
d
P
c
a
n
e
á
s
n
vn
M
a
e
f
x
+
1
A
>
>
+
í
(
a
+
d
=
±
a
X
+
C
f
x
1
A
>
—
f
^
V
^
n
^
W
+
C
w
e
b
e
g
m
l
^
p
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1
20
2.13 BINOMIAL SERIES, GENERAL RELATIONS
Numérica Constants
(a) General case. The nth power
of
a ± x can be expanded by Newton's formula in a power series
called
a binomial series.
(.*»)=«•(. ±)•=«•[. *
(;).+ (2>-
*...
*(>•]=.-¿^y
where n = signed
number,
í I =
binomial
coefficient, a = (±1) , B= (±l)á, and u = x/a.
(b) Classification. If in theexpansión above
n
= 0,1 ,2 ,3 , . . . ,
p
the series is finite
n # 0 ,1 ,2, 3,... ,p anda2 < 1 the series is convergent
n # 0,1,2,3,.. . ,
p
andu2 > 1 the series isdivergent
(c) CAmodel. The most convenient model
of
this series is the nested series introduced in (1.07),
which for« =
x/a
is
(a ± ) = Á
(1
±ku) =a (l ±nu(\ ±í-^—
a(l
±^ . ( l±
))))
Five distinct forms of this mode l are given below in
(d),
and selected particular cases are
tabulatcd in (2.14).
(d) OistinctCAmodels. Uu'2 < I a nd X = u/p,
n ——m,p = 1
1 i tJ i
Tm+
' h
-m
+ 2
(, rm + 3,iT lll\
(1±«)- ,TmTT 2 (,T 3 Ul'T 4 U(T
))))
n —
-, m = 1
P
vTT;= .±a(,
*>-
'*(.
=2% '*('
*3\~ «X(l *-))))
1
n
=
,m
= 1
P
^, ,^(, ,^-(,=^.,( , ,3^. , ,
,..,)))
m
n = —
P
{/(I ±«r - 1±
mA'(l
T>2H .V( i;2í3 A'(l T*4 .V(l T))))
m
1 . T ../. T /> + » . ./ . ^2/>+ mtY. T3/>+ m \ \ \
7¡===ITHa(.T 2 A(lT 3
A(lT
4 A(. 1 )))]
where
m,/>
are positive integers or positive decimal numbers.
(e) Absolute truncatlon error
\e\
in these seriesis lessthan the absolute valué of the first dclctcd
t e rm .
In series (a), |er+l| =S
M Ju,+ l a
In series (c), |fr+1|
<
|c,r2rs
•
cfc,+ ,u'+,\a*
(f) Example. By series (c).
(I 4- 0.234)°* =I+0.567(0.234)(l - ^(0.234)(l - -^p(0.234)(l - — (0.234))))
= 1.126 601 (correct valué 1.126615)
, • . ^^ ,v /0«3 \ /1 .433 \ /2 .433 \ /3 .433 \ _ v.
and |))))))
_i_
=
J:(lTWlT5„(lI|„(1T2„(lT „(lT5„(1T...,))))))
(b)
Exponent
n=1/m (m = 2,3,4;«= x/am)
y/a
±x=Vaíl ±
«(
^a
±x
=
V^(l
±
«(
V a ±x=
Va(l
±
«(
(c)
Exponent
n=
-1/m
(m
Va ix
Va V
1 1 / i
»o»«
1 1 /
WTx=Ta\i:fu[
v£
( H'^-O'H ^' •< '))))))
('H I ( -( t ( >
*-'))))))
(.H-( í ( 7-( T-( > *->))))))
= 2, 3,4; u = x/am)
(.*|.(.*f.(.*H H >
*-'))))))
0*i-( HlT7o-( T-( f *- >))))))
('*H H 7-{ T-( 7' *->))))))
(d) Exponent /» = p/fl (p = 2,3,4;a = 3,4,5;«= x/a?)
iíw^f=.'(.*4 *M1 * -(' t í-(' t7'(: t 7-(i t >))))))
tf7±7P
=
'(>
±
fc(l
í
i.(l
T
.(l
*
|«(l
T
^lífl l í
o))))))
^7±7F=«(1 ±
«.(i
*^(. *f^i í ^ h f(i *£.(.*• )))))))
(e)
Exponent
n
= -p/
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1
1
22 2.15
GENERAL
SERIES AND PRODUCTS
OF BINOMIAL COEFFICIENTS
' 1 ' i
Numérica Constants
(a)Generalsums of complete, monotonic sequences
(n,r,N = 1,2,3,.. .)
ÍQ-Q +OQ +- +
Q-*
l/0=(MM2 )+-+0 , +
('-,>r
| ^K)+
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2
2
1
E
U
S
C
e
N
m
c
C
a
(
a
D
n
o
T
y
m
e
e
h
m
t
=
m
(1
+
m
(
1
4
n
=
7
.
_
m
«
—
o
a
s
h
o
h
u
a
s
y
e
m
o
o
h
m
I
n
C
H
m
e
o
h
e
s
b
h
i
a
o
a
a
a
n
m
(
b
C
c
a
o
O
e
o
e
e
o
a
a
m
s
h
m
o
n
n
e
e
a
n
a
o
h
w
o
w
F
m
e
a
d
o
m
o
r
—
4
€
=
7
4
6
(
<
0
s
1
1
o
d
e
y
b
C
a
1
\
o
+
e
=
2
7
e
(
<
2
3
1
6
e
=
\
I
+
—
)
+
=
7
+
(
e
<
4
x
)
/
1
\
1
O
+
=
7
+
e
(
e
<
4
X
6
/
1
\
(
H
=
+
7
(6
6
°
k
1
(
c
E
e
s
e
b
1
1
I
1
1
c
e
n
T
(
d
E
e
s
n
r
a
o
I
e
=
2
+
I
+
2
+
3
+
4
+
•
2
1
T
a
d
g
o
h
s
e
s
h
w
n
e
o
n
e
<
6
(
e
V
ú
o
e
e
m
a
o
m
o
5
+
6
+
•
c
g
ra
d
y
a
s
h
w
i
n
(
e
(
o
n
1
6
6
X
°
a
s
s
o
C
m
m
S
e
F
o
1
2 3
4 5
2
0
2
6
2
7
2
7
2
7
2
0
2
6
2
7
2
7
1
2
7
6
7
8
9
1
2
7
2
7
2
7
2
7
2
7
—
2
2
7
1
2
7
~
1
~
N
m
c
C
a
(
a
D
n
o
T
y
m
y
e
h
m
v
H
f
+
+
+
-
n
=
5
y
i
Ü
2
3
n
I
N
o
s
y
a
a
e
w
h
y
s
a
a
o
n
m
(
b
C
c
a
o
O
y
o
e
e
o
a
a
m
s
h
h
u
o
h
g
v
m
o
a
g
n
h
w
o
w
o
b
h
n
a
o
s
e
n
(
c
E
m
e
.
+
+
+
+
L
n
1
6
=
5
6
(
e
<
X
0
4
2
3
1
V
+
+
4
4
L
n
1
e
=
5
2
6
(
6
<
7
y
+
3
1
,
+
+
+
+
L
n
1
-
6
=
5
6
(
6
<
°
2
1
E
U
S
C
y
2
(
d
E
e
s
n
f
r
a
o
l
(e
V
ú
o
y
n
e
m
o
m
o
y
=
0
+
n
F
o
1
2
3
4
5
1
0
0
5
0
6
0
5
0
5
6
7
8 9
1
0
5
0
5
0
5
1
0
5
0
5
I
0
5
1
+
1
+
1
2
+
1
2
+
1
+
4
+
3
+
1
+
5
+
c
g
v
y
r
a
d
y
h
w
i
n
{
(
o
n
1
e
<
5
X
°
a
s
s
o
C
m
(
S
m
e
a
d
o
m
o
b
m
e
a
o
e
h
g
s
o
d
f
u
y
=
—
-
6
5
3
6
(
6
8
O
3
1
—
-
6
5
7
6
(
6
1
x
5
1
—
+
5
+
(
6
8
7
.
3
a
o
o
n
c
a
o
y
=
I
_
=
5
e
(
e
<
3
x
4
i
(
g
F
b
C
S
o
m
a
n
e
m
O
R
=
1
+
a
=
7
8
2
1
)
i
s
h
a
o
m
o
a
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~1
^
20 2.20 NATURAL LOGARITHM OF 2 Numerical Constants
(a) Definition.
The symbol ln2 denotes the natural logarithm of2,
defined
bythe relationship
e,a2 =
flim ('+-)] whcrc
m2 =
°-693
147
180559 945
309417
232
This constant is an important component in the construction ofCA modcls of natural logarithms
of large numbers (9.06).
(b) Calculation Of ln2 to a desired degree of aecuracy can be accomplished by
means
of continued
fractions and infinite series as shown below. For simple and rapid calculations,
ln2= Vi - 6 = 0.693 375 245 - 6 (6 < 2.3 x
I0 4)
(c)
Napier's
continued fraction,
1
ln 2 = 0 +
1 +
1
2 +
3 +
1 +
6 +
3 +
1 +
1 +
2 +
converges
as
shown
in
(/ )
(for
n
=
11, \e\ <
2.8x
10~a).
Simple and rapid
approximations
by
simplefractions are the convergents of this fraction constructedby (2.08/).
(d)
Gauss' series
based on
for
x
= g yi eld s
I
- = - + — +
—7
+
1 x 3*J 5* '
ln2
=T+_+_ +
.. .
where
a
= 5.
The
relatively slow con
vergence of this seriesis shown in ( /)
(forn = II, 6 <
4.1
X 10-7).
(e) Goursat's series basedon
N+
I = f l 1
N 12N + I 3(2N + l)3
' ]
(2^+
I)'*
for N = yields
• r¿ t>* f 1
where
b
^-4^-The better convergence^of
this serie;* is
shown
in
(f )
(for
n
= 8,
6 < 5.1 :; lO '0).
(f) Valúesof ln2 based on n-termapproximation
i
Fraction
Gauss
Goursat
1
1 000000000 0 600000000
0 666666667
2
0 666666667
0 672000000 0 691358025
j
0 700000000
0.687 552000 0.693004 115
4 0.692307692 0.691551 016
0.693 134757
5 0.693 181818
0 692670830
0.693
146047
6
0.693 140794
0 693000646
0.693 147074
;
0 693150685 0.693011 129
0.693 147 170
8
0 693146417 0 693132459 0.693 147 180
9
0.693 147362
0.693 142 154
10
0 693147097 0.693 146226
11
0.693147208
0.693 146672
ñ
— 1
1
Numérica
Constants 2.21 FIB0NACCI
NUMBERS 29
(a)
FibonaCCi numbers
Fr
(r= 0,
1,2,
3,...)
are defined
by
their
gencrating rclatton
Fr = /%-, + /= ,-
where
F0
= 0,
Ft
= 1, F2 = 1,
Fs
= 2,
F4
= 3,
numbers are given belowin (*)
(b)
Closed form ofthis number is
Numerical valúes of the f irst 50 Fibonacci
r . -u^ -^y-^-
where
a = 1.618033988 749894848204
are the gotden mean
numbers.
(c)
Golden mean numbers arethe roots of
x*
- x -
1 = 0
which is thecondition of thedivisión ofa linesegment AB in
the
adjacent
figure by the point C in the mean
and exteme
ratio,
so that
the greater
segment
AC
is the
geometric mean
of
AB
and
CB,
B
=
-0.618
033
988
749
894 948204
* = V(i)(* + i)
and the
golden rectangle
is the rcctangle
whose
sides have the
ratio
ofé(l
+VÜ)
and
is
supposcd to
have the most pleasing cffect on the cyc.
(d)
Series representation
of the u + l)th
Fibonacci number
is
'--¿(,1*)
where m is the n ear es t higher i nt eg er
above j ror |r.
(f)
Ratio of two consecutive Fibonacci
numbers
is
K
=
FJF,.
where R, =
1 +
I +
1 +
1 +
1 +
is a continued fraction with r divisions.
(g) Examples
By (b),
FB ^yjrí 8 -
0 )
=21-
(e)
Table of Fibonacci numbers
r
F,
r
Fr
r
F,
1
1
18
2 584
35
9227
465
?
1
19
4 181
36
14930 352
3
2
20
6 765
37
24157817
4
3
21
10946
38
39088 169
•i
5
22
17711
39
63245986
6
8
23
28657
40
102334 155
7
13
24
46368
41
165 580141
8- 21
25
75025
42
267914 296
9
34
26
121393
43
433494437
10
55 27
196418
44
701408733
11 89
28
317811
45
1 134903 170
12
144
29
514229
46
1836311903
13
233
30
832040
47
2971215073
14
377
31
1346 269
48
4807
742049
15
610
32
2178
309 49
7778 742049
16
987
33
3
524578
50
12586269025
17
1597
34
5 702 887
-twí *~t>®*GVQ-«
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~ ~J ~J 1 H
'
H
30 2.22 BERNOULLI NUMBERS
(a) Generatlngfunction
e'
- 1 fT0
'r
O 1 2 3
where
(W < 2jt)
Numerical
Constants
B0 =
1
¿ 2 = é
a - L
a4 311
*« = é
« 30
« = -i
*>IO «i
i , = -é
¿ 3= 0
¿s = 0
¿7 = 0 fi„ = 0
¿M=0
are Bemoulli numbers of order r = 0, 1, 2, (A. 10).
(b) Auxiliary generatlng function
„
x x ^ n
x2' B„
Btx2 B.x' B^x6
2- í - í ,?„Bw=«+ ir+
i r+ ir+
where
(M <
*)
B0= \
fl,
^*I
_30
*»> — 42 4 — 30 f. — 60
are
auxiliary
Bemoulli
numbers of
or de r r = 0, 1.
2. . . .
(A. 10).
(c) Relatíonships. Bcrnoulli numbers Br and
B,
oforderr canbe alwayscomputed in terms oftheir
lower countcrparts as
r - 1 1 A
ír+ l\ -
B'=i(7T0-rTT.?,(2*K (-4,6,8....,
r-
l ¿
„/2r
+
l\
2r - 1 l ¿ „, 2r + l
5 '=^
=ai(i
wherej = ±r - 1, / = r - I, a =
(-1)'+',
and 8 =
(-l)í+\
(d)
Examples.
If B2 = ¿, thenby (c),
s 3 1 /5 \ - 1
,-
1
B. =
( )B2
=
B,= (-l)iB.
= + —
4
10 5\2/
2 30 - V 4 30
Similarly, if B2 = ¿
ana
¿4 = —
o> (hen again
by (c),
(e) Series representation. For r =
2,3,4, . . .
and a =
( - l)'*',
- „ (2r) / I 1 1
)
The series converges rapidly for r > 3.
(f) Example. From
(b), Bs
= ¿ = 0.075758,and bya seriesof three terms,
*5 =
¿w
=2(2^[I +
¥°
+3^] =075758
' 1
Numerical
Constants
I
2.23 BERNOULLI POLYNOMIALS di
(a)
Definition. Bemoulli polynomials
B,(x) oforder r=0,1,2, 3,... are defined
as
*w=«*+(;)»-¿.
Q>-,s>+O' 15»+•
+0*
where B0,B,,B2,... ,B, are the Bemoulli
numbers introduced
in
(2.22a).
Particular
polynomials
for r = 0 ,
1,2,3,.. . .
25are
tabulatcd
in (A.l1).
(b) PropertieS. In general,
B,(x
+
1)
-
B,(x)
= «'-' Br(x) = -a# ,( l - x)
Forx = 0,r= 1,2, 3,..., the polynomials define the respective Bcrnoulli numbers as
¿2,(0)
=
K =
ccB,
¿2,+i(0)
=
B2t+l
=0
wherea=
(-l)'+l.
(c) Derivatives of B,(x) are
dBt(x)
dx
=
rB,_t(x)
£M±=r(r-l)B,_2(x)
dx
and in general,
'
Qnl
BK.m(x) n 0
(« = r)
( » > 0
where
Dn
= — and «=1.2,3,
dx
(d) GraphS
of the f irs t four
polynomials are
shown
below.
(e) Indefinite
integráis
of B,(x) are
I Br(x)
dx
=-^-¡-+
C,
jj/írw«fflf
(r+l)(r
+
2)
andin general,
»,5
/rU
_
£
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1
32 2.24
EULER
NUMBERS
Numerica l Constants
(a) Generating function
i ^=v l> ' =£o +
i1 i
+ ¿1£ +£¿ +...
(|x|
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-— i 1
—I •— 1
34
2.26 RIEMAN ZETA FUNCTION
1 1 i — 1
Numérica Constants
(a) Definition. Rieman zeta
functions
Z( r) of o rd er r =
1,2,
3, . . . are defined by the series
A I 1 I I
zw=.?,P =P+?+?+
or bythe product
«n-ní-i) --
where the product is taken over all prime numbers
p
= 2, 3,5, 7, 11, 13,
(b) Alternativeseries definingZ(r) is
2' A
I
2' /
I
1 1 \
z(r) = y = ( - + - + - +
2' -
iiT,(2*
- i)' 2' - i
\v y y
/
The series converges rapidly for r > 5.
(c) Special
valúes of Z(r) are
Z(,) =
oo,Z(2)=^,Z(4)=^,Z(6)
=^....
and ingeneral, in tcrms of Bemoull i numbers and polynomials (2.22), (2.23),
{2n)'\B,\
+
o- if
r is
even
Z(0 = 2
T,
=
i-i
*
(2*)' f• B,(x)
-.fifi/'
r Jo
(cot-T.v ¿r i frisodd
where r > 1and a = (-l)'+l. Numericalvalúes ofZ(r) for r = 1,2,3,. . . ,20 are tabulated in
(2.27r). Large tables ofZ(r), including decimal valúes of argument, can be found in Ref. 2.03.
(d) Sums of reciprocal powers
i l i l í
i_,* 12 3 4
l
l i l i
/ \'
X = - + - + - +
-+-=(-Z
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~§
1
36 2.28 GAMMA FUNCTION, GENERAL RELATIONS
(a) Euler's integral. Gamma function ü(x + 1)is the generalizaron
of the factorial (2.09o) and is defined as
r(x+
I) = I
e-t'dt = x\ (*>0)
(b)Gauss' limit is a broader definition of the gamma function
expressed as
T{x
+ 1) = lim
—-(x + n){x + n - 1) - - - {x + 2)(x + 1)
-jáL-TiC-T1) -
Numerica l
Constants
iru+i)
The graph
shows
that
T(x
+ 1) is
single-valued except
at
x = - I, -2 , -3, . . .
, and
its alternating extremes are T(l.462) =
0.886,
T(-0.504) = -3.545, T(-1.573) =
2.302,
r(-2.6ll)
= -0.888, T(-3.635) =
0.245.
T(-4.653)= -0.053, T(-5.667) =
0.009,
T(-6.678)
=
-0.001
(Reí. 2.08).
(C)
Functional equations. With
the restrictions
placed
upon
x
in
(b),
r(x
+ n) =
(x
+
n
- I)(* + n - 2) u +
2)(x
+ l)r(* + 1)
rt.-.)-7
0i±Jl
(* - «)(* - n + I) (x - 2)(* - 1)*
wherc
n =
positive
integer and
x
=
signed
number (positive or negative,
integer
or fraction).
(d) Integer and
fraction
arguments (n,p,q = l. 2.3....)
r( „ + i)
= n(n - )( « - 2) 3 -2 - I = ni
= (n - 2)(n - 3) 3 2 1 =
(n •
-2)
r(n + p + 1) = (n + p)\
r (n -p+
) = («
-p ) \
rK)=K-0K->)-
K)H)K>H)
(e) Reflectlonand duplication formulas
n
r(«)r(-«)
=
u sin uJ T
where
x
= signed number, « =
1,2,3 , . . . ,
and
u <
1
( f) Special valúes (n = l, 2,3,.. .)
4*
r(2*)=^=r(*)r(*
+
2-)
r » r ( i
- „)
r ( -n)
= *
f
(o) = *
r( i) = i
£(n) =
(n -
- D
n-k)
=
-2 \£
rtí) = V^r
n-+
)
- (2n 4^
4
r(-« +
A)
=
(_4)--'̂
(2n)
Numerical valúes ofT(n + 1)
and
r(u)
are
tabulatcd in (A.03)
and
(A.04),
rcspcctively.
NumericalConstants
1
~i
H ' 1 1 1
2.29
GAMMA
FUNCTION, NUMERICAL
EVALUATION
37
(a) Smallargument. For o < u ^ 3, the gamma func- (b)
Factors
akand¿*
t ion can be evaluated to a dcsi red degrec ofaecuracy
. / (1 - u)un
, / (1 +
u)u3t
ni - u = rrW--i—-?— +
(I
—«)sm«jr
where in terms of ak given in
(b),
z = atu —OjU3 —a5«5 — —fl|j«
and \e\ S 5 X 10 '°. Nestedform intcrmsof Ak given in (A) is
r =
A,u(l - i4,«*(l
-
.4s«-(l
-
A,u2(l A,3u2))))
The
general expressions
for ak and
Ak
are a, = I - y,
a:l
= Z(3)—I,
a5
= Z(5)— I,... and
A,
= a, ,
A3
= a3/a,,
i45
=
as/a3,...,
where y = Euler's constant
(2.19)
and Z(r) = Riemann
zeta function (2.26).
(C)
Example. If
T(1.25), T(0.75), T(3.25), and
T(3.75)
are dcsired,
eT
is calculated first with
u = 0.25.
k
«1
Ak
1
0.422
784335
0.422
784335
3
0.067352
301 0 . 159306581
5 0.007 38 5 551
0 . 109655511
7
0 .001 192 754
0.161498309
9 0 . 000223
155
0.187092229
II
0.000044926
0.201
321951
13 0 . 000009439
0.284 490050
axu = 0.105 696083 8
-a3«3=
-0.0010323797
the sum of which is r = 0.104
636418
2 and eT = 1.110306 850. Then by relalions (a),
-as i r
=
-0.000007
2124
-a7«7 = -0.000 000 072 7
,.* —
-a.,u = -0.0000000008
n,
, n„A r
/(0.75)(0.25)tt
/(L25)(0.25)g
r(,+0-25)-íV,.25sin0.25^ T(l -
0.25)
- , V0.75sin0.25^
0 . 906 402
47 7 3 + e
1.225 41 6 702 2 + 6
where€ = -2 X 10 ' basedon comparisonwith the tabulated valúesin (A.04). The remaining
valúes are calculated by (2.28a*).
(d)
Large
argument.
If
x > 4
isan integer or
decimal
number, the Stirling's
expansión
(2.09í),
r{x + i)
tl{^'y/2xH
+ € = xl
yieldsgood rcsults as shown in (í) below.
(e)
Errors
of Stirling'sexpansión of r(x + 1)
X
+ 1
Correa valúe o(T(x
+ 1)
Approximatc valúeof ü(x +
1)
3.00
2.000000000
(+00)
1.999997
800
(+00)
3.25
2.549 2 56 96 7 ( +0 0)
2.549 2 55 92 8 ( +0 0)
3.50
3.323
3 50 97 1 ( +0 0)
3.323
350
418
(+00)
3.75
4.422988 410 (+00)
4.422988086 (+00)
4.00 ,
6.000000000
(+00)
5.999999
792
(+00)
5.00
2.400000000 (+01)
2.399999993
(+01)
6.00
1.200000000 (+02)
1.200000000
(+02)
7.00
7.200000000 (+02)
7.199999 999 (+02)
1ÜM) i
5.040000000 (+03)
. 040000000 (+03)
10.00 ' 3.628800000
(+05)
3.628800000 (+05)
20.00
1 .2 1 64 51 0 04 (+17 )
1.216451004
(+17)
50.00
6.082 8 18 6 40 ( +6 2)
6.082818 640 (+62)
__^^
-
8/9/2019 Handbook of Numerical Calculations in Engineering
23/206
3
2
3
O
O
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B
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=
*
=
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I
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(
x
*
*
r
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+
)
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+
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o
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p
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1
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T
x
1
N
m
c
C
a
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A
(
=
x
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+
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r
u
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^
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n
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a
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n
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x
J
x
B
«
+
h
x
r
(
2
)
s
n
T
n
+
>
x
=
J
<
T
2
1
B
n
+
+
n
+
x
r
v
l
(
2
c
J
x
w
e
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=
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2
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a
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é
a
«
H
-
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H
-
8
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«
B
u
1
u
=
n
u
s
n
«
(
d
n
e
a
r
a
o
a
g
m
s
»
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=
1
2
3
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(
>
h
a
r
>
»
<
—
M
r
^
~
M
l
B
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+
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+
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+
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2
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+
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2
-
2
2
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^
1
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2
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2
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2
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»
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¿
A
-
8/9/2019 Handbook of Numerical Calculations in Engineering
24/206
~~Jj 1 ' 3 ' 1 ' 1 1
40 2.32
DIGAMMA
FUNCTION, GENERAL RELATIONS
(a) Derivative of natural logarithm of T(x + 11 in (2.28a) is called
the
digamma function rp(x).
„,).¿{nn,ti)»-Efe+Jl
where T'(x + 1)is the derivat iveof T(x +
11
with respect to x.
(b)
Natural logarithm
ofGauss'limit in
(2.296)
canbe
diQcrcntiatcd
term by term so that
1 ~H
1
Numerical
Constants
M =im (in. - i j^) =i
(1
- --^)- y
where y = Euler's constant (2.19a). In some literatures
ip{x)
is calledthe
psifunction.
The graph
in (a) shows that \p(x) issinglc-valuedexcept at x = —1,—2,—3,... , and its zero pointsareat
x = 0.462,
-1.504,
-2.573,
-3.611,
-4.635.
-5.653, -6.667,
-7 .678 , . . . .
(c) Functional equations with restrictions placed upon x in
(b)
are
1
x¡){x +r )= x¡>(x) + 2
a^i
x
+
k
where r ¡sa positive integer and x is a signed number.
(d) Integerand fractionarguments (n,p,q
=
1.2.3—)
V v - r) =
Y(Jt)
- 2
* f
, * + 1 - *
*(.) = +j +5 +-- + ,_ l +
1 ¿ 1
;-r-.?.¡-r
*KH(*)+4.,+'?*
v1,.í)=v(.í)+,¿ '
a/
\ q)
k.,qk
- ¿
(e) Reflectionandduplicationformulas
V(x
+1-
n)
- lfi(x
+
I)
=¿
.i x — k
tp(x) - y>(x+ l) =
2y(2x - 2) = y( * - O + V(* - 5) + 1 4
y(«)
- V(-«)
where x = signednumber, n = 1,2,
3,. . .
. and | a| < 1.
(f) Special valúes
u tan
uJ l
V»(-n) = ±» V(0) = -y V(D = l ~ 7 V(2) = 1+ 5 - Y VO) = l+ | + y
v,(_i) = _v_|n4
V(-5)= -y-lnV27 +
V3-
V(~¿) =
- y - ln 16 + -
' 2
V(j) = 2 - y -
ln 4
V(5)
= 3 -y - l nV27 -
VÍJt
V(í) = 4-y-lnl6--
Numerical valúes ofV(u)
are
tabulated in
1
-
8/9/2019 Handbook of Numerical Calculations in Engineering
25/206
'
4
(
V
VWC
w
(
a
—
j
Á
V
T
m
vm
t
(W
(m
m
(
m
VWmmm
w
V
j
.m
y
m
w
(m
vW
s
y
r
w
(
u
y
y
V
^
t
V
N
N
OO
(m
rm
;
4
=
aH
-<
a
—
1
a
d
—
a
r
2
(
1
—
A^
d
1
*—AV
d
asm
—
d
d
-
M
w
(
d
•
D
•
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
(d
a2
u5«
u
—^
d
v
l
2
T
d
-
8/9/2019 Handbook of Numerical Calculations in Engineering
26/206
J
4
2
3
P
O
G
M
F
U
O
N
M
E
C
E
U
O
(
C
n
(
e
I
n
e
m
a
e
g
m
F
§
<
<
h
e
c
o
m
a
n
(
2
3
y
e
d
r
e
u
a
N
m
c
C
a
x
X
r
+
=
V
(
2
1
(
*
-
w
e
r
=
2
3
V
(
«
V
»
a
c
c
c
a
e
b
(
2
3
a
t
»
)
«
Í
>
m
u
a
e
o
a
n
2
3
n
e
m
o
h
e
n
ú
*
>
«
V
,
«
+
2
T
_
(
*
—
U
^
«
=
«
+
3
(
E
m
e
I
h
ú
o
V
3
5
a
y
M
0
5
a
c
e
h
e
y
(
3
+
5
=
V
(
0
5
p
—
~
Y
2
2
a
b
23
*
<
0
5
=
—
U
+
-
V
°
5
=
9
7
(
0
5
s
2
(
g
L
g
g
m
v
>
s
i
n
e
o
d
m
n
m
h
v
v
o
S
n
s
ó
i
n
2
3
y
e
d
a
n
a
w
h
=
(
—
1
.
1
I
<
>
a
w
-
1
<
w
e
¿
=
L
+
L
+
<
x
6
3
4
7
3
6
r
>
1
_
J
_
3
5
d
~
2
+
6
v
+
6
d
m
_
r
w
+
_
w
+
.
m
+
_
m
+
5
w
+
M
<
*
1
V
I
2
3
*
2
5
*
2
7
a
6
9
a
u
I
v
e
m
o
h
y
m
o
c
e
e
n
u
n
o
o
o
1
d
g
d
s
p
a
T
c
A
b
o
w
s
h
w
h
a
o
a
c
o
o
h
s
o
m
a
o
T
v
(
h
E
o
o
S
r
n
s
ó
(
1
(
*
X
C
e
u
y
*
A
o
m
c
ú
o
V
x
-
0
5
0 0
5
1 1
5
a
a
2
=
4
9
(
+
„
C
=
1
6
(
+
j
2
=
3
(
0
„
%
-
=
4
(
0
.
/
2
4
1
1
3
)
=
4
9
(
0
N
a
c
e
f
o
£
1
5
3
;
6
(
1
1
2
)
=
9
(
0
J
T
6
-
1
I
2
+
1
3
)
=
2
8
(
0
3
9
(
0
2
8
(
0
2
2
(
0
j
r
6
1
+
2
+
3
+
=
2
(
0
5
K
j
6
1
+
2
3
+
4
+
5
=
1
8
(
0
.
7
1
1
2
+
1
3
-
1
1
-
=
9
5
(
0
1
8
(
0
9
5
(
0
3
N
M
R
C
A
D
F
E
R
N
-
8/9/2019 Handbook of Numerical Calculations in Engineering
27/206
m 31
46
3.01 FINITEDIFFERENCES AND CENTRAL MEANS
(b) Divided
difference
of first
order defined as
f{Xfxk+l) — _
, _
x
xk*\ *k ** + l **
isthe
slope
ofthechord
k,k +
I,
ta n q> =
where J t= 0, ±1, ± 2,
±3
(c) Forward difference of first order, defined
as
a*í+i)
--/*(**)
=****
-y *
is thedifference of the valúesof the function at xt+l, xk,
d cnot ed as
Ax, =yk+\
-j k
(d) Backward difference
of
first order, defined
as
A**)
-A**- i ) °x -J*- '
is thedifference of the valúesof the funedon at
xk,
xt_,,
denoted
as
v>*
= yk - jk-i
(e) Central difference
of
first order, defined as
A*.+in)
-/(**-i/a)
=yk+\n ~ Jk-m
is the
difference
of the valúes of the functionat
xk+m,
xk-in> dcnoted as
fyk =7*+1«~ yk-
where yk.m, yk+t/2 are the vertícal coordínales of
midpoints
betwccn k - 1, kand *,k+
1. respectively.
(f) Central
means
of first order,
defined
as
f[xt-in) +f(xk+,n) yk-xn
+yk
+m
2 2
Numerical
Differences
(a)