Handbook of Numerical Calculations in Engineering

8/9/2019 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


4/206

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5/206

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


6/206

— 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)


7/206

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).


8/206

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)


9/206


10/206

1

2

P

O

N

B

C

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(

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f

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+

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*

M

M

-

*

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11/206

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


12/206

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


13/206

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


14/206

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/


15/206

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)+


16/206


17/206

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


18/206

~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-«


19/206

~ ~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

_

£


20/206

1

32 2.24

EULER

NUMBERS

Numerica l Constants

(a) Generating function

i ^=v l> ' =£o +

i1 i

+ ¿1£ +£¿ +...

(|x|


21/206

-— 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


22/206

~§

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)

__^^


23/206

3

2

3

O

O

W

T

H

G

M

F

U

O

(

a

B

n

m

a

c

c

e

1

e

n

e

m

o

g

m

u

o

(

*

=

*

=

H

I

W

(

x

*

*

r

x

k

+

)

r

k

+

w

e

k

0

1

2

a

s

g

m

e

—

—

—

3

(

b

F

o

a

p

y

m

a

o

s

e

2

1

a

o

o

d

m

n

s

T

x

1

N

m

c

C

a

s

A

(

=

x

x

l

x

-

2

x

m

+

1

=

1

r

u

m

+

i

r

*

+

i

x

(

x

l

x

2

x

m

-

l

x

m

T

x

m

+

1

w

e

m

=

p

v

—

=

—

a

s

h

g

m

e

c

e

2

2

(

c

F

o

a

p

y

m

a

o

s

e

o

d

m

n

s

A

>

x

k

x

2

A

x

(

m

-

D

=

(H

A

(

-

+

)

(

H

.

r

=

(

x

A

x

2

A

x

m

r

^

+

m

)

w

e

x

e

h

n

i

n

b

a

h

s

v

a

(

d

D

e

a

o

a

d

e

a

x

i

s

p

a

c

a

f

a

o

a

p

y

m

a

w

h

x

2

x

2

1

A

2

a

n

1

2

3

I

n

c

a

(

2

ü

2

2

2

2

4

6

4

2

2

n

=

2

T

n

1

2

>

-

\

2

T

+

\

(

2

1

1

=

(

2

1

2

3

2

5

5

3

1

V

\

(

e

N

u

a

o

h

m

h

m

u

o

o

0

u

;

s

V

1

—

u

u

.

.

.

—

i

—

—

+

a

u

a

U

a

-

a

(

1

u

n

/

(

1

u

u

.

,

.

l

n

T

a

=

n

W

—

a

u

«

+

«

+

+

1

—

w

n

w

e

a

a

a

e

h

a

o

s

e

n

2

F

1

(

«

«

<

n

2

3

l

n

n

u

=

l

n

n

u

I

+

l

n

n

u

2

+

+

l

n

1

u

+

l

n

T

+

u

w

e

n

T

+

u

s

h

c

a

o

(

S

r

n

s

ó

o

n

E

x

+

o

v

s

_

n

r

*

+

=

n

+

n

V

+

w

e

A

s

h

e

n

2

0

a

f

o

v

y

a

g

A

0

N

m

c

C

a

s

2

3

P

A

B

U

O

3

G

R

O

(

a

P

u

o

í

x

s

h

g

a

z

o

h

a

o

a

2

0

a

s

d

n

n

x

=

x

+

1

=

t

d

=

(

x

>

w

e

s

h

a

a

o

a

A

r

e

a

o

n

o

n

(

2

2

o

(

2

3

a

e

c

e

n

o

a

o

w

h

h

s

u

o

(

b

B

a

u

o

H

x

y

s

d

n

B

*

r

=

-

(

-

0

d

=

(

+

<

>

>

>

a

s

e

a

e

o

h

m

u

o

u

o

B

x

r

H

^

r

x

B

x

y

=

n

x

\

U

y

1

n

x

-

I

(

c

R

e

o

o

m

a

(

n

=

1

2

3

)

B

x

y

=

B

r

x

B

x

y

z

=

B

¿

x

+

B

x

7

B

x

+

z

=

B

*c

B

+

x

_

w

+

v

n

-

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

P

=

/

»

=

V

A

-

x

2

-

x

P

=

x

2

x

2

x

&

=

e

=

m

%

&

=

5

a

*

-

*

]

Q

é

a

«

H

-

*

H

-

8

F

«

B

u

1

u

=

n

u

s

n

«

(

d

n

e

a

r

a

o

a

g

m

s

»

?

=

1

2

3

)

(

>

h

a

r

>

»

<

—

M

r

^

~

M

l

B

>

+

/

+

2

+

I

2

>

+

2

>

2

-

2

2

>

I

w

#

^

1

B

^

+

I

2

_

2

>

)

2

-

2

2

3

+̂

B

»

>

B

¿

A


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


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


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


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)

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