chapter 6 gamma function and related functions

6. Gamma Function and Related Functions

PHILIP J. DAVIS

Con tents

Mathematical Properties. . . . . . . . . . . . . . . . . . . .

6.1. Gamma Function. . . . . . . . . . . . . . . . . . . .

6.2. Beta Function . . . . . . . . . . . . . . . . . . . . .

6.3. Psi (Digamma) Function. . . . . . . . . . . . . . . . . 6.4. Polygamma Functions. . . . . . . . . . . . . . . . . . 6.5. Incomplete Gamma Function. . . . . . . . . . . . . . . 6.6. Incomplete Beta Function. . . . . . . . . . . . . . . .

Numerical Methods . . . . . . . . . . . . . . . . . . . . . .

6.7. Use and Extension of the Tables. . . . . . . . . . . . . 6.8. Summation of Rational Series by Means of Polygamma Func-

tions. . . . . . . . . . . . . . . . . . . . . . . . .

References. . . . . . , . . . . . . . , . . . . . . . . . . . .

Table 6.1. Gamma, Digamma and Trigamma Functions (1 5 s l 2 ) . .

r(x), ~n r(x), +(z), +'(z), ~=i(.oo5)2, IOD

Table 6.2. Tetragamma and Pentagamma Functions (1 5 x 5 2 ) . . .

+"(x), $J3'(2), ~=1(.01)2, 1OD

Table 6.3. Gamma and Digamma Functions for Integer and Half- Integer Values ( l l n 5 1 0 1 ) . . . . . . . . . . . . . . . . . .

r(n), 11s +(n), IOD

l/r(n), 9s n!/[(2?r)h"+3]~", 8D

r(n+$), 8s Inn-+@), 8D

n=1(1.)101

Table 6.4. Logarithms of the Gamma Function (1 I n 5 101). . . . . .

loglo r(n>, 8s log10 r(n+$), 8s

log10 r(n+#), 8 s In r(n)-(n-$) lnn+n, 8D

log10 r(n+$), 8s

n= l(1) 101

Page

255

255 258 258 260 260 263

263

263

2 64

265

267

27 1

272

274

National Bureau of Standards.

253

254 GAMMA FUNCTION AND RELATED FUNCTIONS

Page Table 6.5. Auxiliary Functions for Gamma and Digamma Func-

Table 6.6. Factorials for Large Arguments (1005nS 1000) . . .

n!, n= 100(100) 1000, 20s

Table 6.7. Gamma Function for Complex Arguments. . . . . .

In r(z+iy), 2=1(.1)2, y=0(.1)10, 12D

Table 6.8. Digamma Function for Complex Arguments . . . . .

+(z+iy), 2=1(.1)2, y=0(.1)10, 5D

%‘+U+iy), 10D B’+(l+iy)-ln y, y l= . l l (-.Ol)O, 8D

. . 276

. . 276

. . 277

. . 288

The author acknowledges the assistance of Mary Orr in the preparation and checking of the tables; and the assistance of Patricia Farrant in checking the formulas.

=.57721 56649. . . Y is known as Euler's constant and is given to 25 decimal places in chapter 1. r(z) is single valued

for the points z=-n(n=O, 1, 2, . . . ) where it possesses simple poles with residue (- 1) "/n!. Its reciprocal i/r (z) is an entire function possessing simple zeros at the points z= -n(n=O, 1, 2, . . .).

Hankel's Contour Integral

and analytic over the entire complex plane, save

(kl< =) 6.1.4 -=-s 1 i (-t)-'e-'dt

r(z) 2, c

The path of integration C starts at + QD on the real axis, circles the origin in the counterclockwise direction and returns to the starting point.

Factorial and II Notations

6.1.5 n(z)=z!=r(z+i)

Integer Values

6.1.6

6.1.7

r(n+1)=1.2.3 . . . (n-l)n=n!

lim -=o= 1 1 (n=O, 1, 2, . . .) z+,, r(-z) (-n-l)!

Fractional Values 6.1.8 I-()) = 2 s m e-12dt=&=1.77245 38509 . . . =(-3)!

0

FIGURE 6.1. Gumma function. *

, y-r(z), - - - - , Y=l/r(4

6.1.9 r(3/2)=$,*=.8~622 692%. . . =(3)!

6.1.10 r (n+ 3) = r(t> 1.5.9.13. . . (4n-3) 4"

r(+)=3.62560 99082. . .

r (4) 1.4-7.10. . . (3n-2) 3" 6.1.11 r(n+#)=

r($)=2.67893 85347 . . .

r (3)

r(3)

1-3-5-7.. . (2n-1) 2"

2.5.8-11.. . (3n-1) 3"

6.1.12 r(n+$) =

6.1.13 l"(n+#)=

r(#)=i.3aii 79394. . .

Ut) 3.7.11.15. . . (4n-1) 4" 6.1.14 r(n+i)=

r($)=i.22541 67024 . . .

*See page 11. 255


Recurrence Formulas

6.1.15 r(z+i)=Zr(Z)=Z!=Z(Z-i)!

r(n+~)=(n-il+z)(n-2+~) . . . (i+z)r(i+z)

= (n- 1+ z)! =(n-l+z)(n-2+2). . . (l+z)z!

6.1.16

Reflection Formula

6.1.17 r(z>r(i-z)=-zr(-z)r(z)=t a c 7rz - tz-1 0 l+ t

=J=, - dt (O< 9 2 < 1)

Duplication Formula

6.1.18 r(2~)=(2~):+ 222-3 r(z) r(z++)

6.1.19 r(3z)= ( 2 ~ ) -1 35’4 r (2) r (z+# r(z++) Triplication Formula

Gauss’ Multiplication Formula

Binomial Coefficient

Pochhammer’s Symbol 6.1.22

@lo= 1, (2),=2(2+1)(2+2) . . . (z+n-l)=- r(z+n)

r (2) Gamma Function in the Complex Plane

6.1.23 r@)=r); In r(Z)=In r(z)

6.1.24 arg r(z+l)=arg r(z)+arctan X

6.1.29 r(i~)r(-iy)=ir(i~)iz= 9r y sinh 7ry

6.1.31 r (1 + iy) r (1 - iy) = I r (1 + iY) (L “?/ sinh ry

Power Series 6.1.33

In r (1 + z) = -In (1 + z) + z (1 -7)

+5 ( - ~ ) ~ ~ ~ ( ~ ~ - ~ l ~ ” / ~ (14<2> n-2

{(n) is the Riemann Zeta Function (see chapter 23).

6.1.34

k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Series Expansion * for 1 /r (2)

ck

1.00000 00000 000000 0.57721 56649 015329

-0.65587 80715 202538 -0.04200 26350 340952 0. 16653 86113 822915

-0.04219 77345 555443 -0.00962 19715 278770 0.00721 89432 466630

-0.00116 51675 918591 -0.00021 52416 741149 0.00012 80502 823882

-0.00002 01348 547807 -0.00000 12504 934821 0.00000 11330 272320

-0.00000 02056 338417 0.00000 00061 160950 0.00000 00050 020075

-0.00000 00011 812746 0.00000 00001 043427 0.00000 00000 077823

-0.00000 00000 036968 0.00000 00000 005100

-0.00000 00000 000206 -0.00000 00000 000054 0.00000 00000 000014 0.00000 00000 000001

2 The coefficients ck are from H. T. Davis, Tables of higher mathematical functions, 2 vols., Principia Press, Bloomington, Ind., 1933, 1935 (with permission) ; with corrections due to H. E. Salzer.

GAMMA FUNCTION AND RELATED F"CX'I0NS 257 Polynomial Approximations'

6.1.35 0 1 x 5 1

r (x+ 1) =z! = 1 + -alx+ -a& + 62 + ag4+ -a&+ E(Z)

55x10-6

~ l = - . 57486 46 ~ 4 = ,42455 49 Uz= .95123 63 U6=-. 10106 78 ~ 3 = - .69985 88

bl=-. 57719 1652 bs=-. 75670 4078 bs= .98820 5891 be= .48219 9394 b3= -. 89705 6937 b7= -. 19352 7818 bq= ,91820 6857 bs= .03586 8343

Stirling's Formula

6.1.37

Asymptotic Formulas

6.1.39

r(-az+b) -l/Z;;e-(U(-az)(U+b--t (la% 4<a, -a>()>

6.1.M

In r(z) -(z-+) In z-z++ I n ( 2 ~ )

(z+m in larg z~<T) B 2 m m +z 2rn(2m-1)2*-'

For B, see chapter 23 6.1.41

1 1 In r(z) -(z-& In z-z++ In (%)+--- 122 360z3

From C. Hmtings, Jr., Approximations for digital computers, Princeton Univ. Press, Princeton, N.J., 1955 (with permission).

Error Term for Asymptotic Expansion

6.1.42

If

R,(z)= In r (z)-(z-# In z+z-+ In ( 2 ~ )

B2m -5 ,,12rn(2rn--l)z an--l

then

where

K(z)=upper U l O bound(z2/(u2+z3) I

For z real and positive, R, is less in absolute value than the first term neglected and has the same i gn .

6.1.43

9th r(iy)=%?ln r(-iy)

-& In (2.) -w-+ln y, (y++

6.1.44

A n r(iy)=arg r(iy)=-arg r(-iy) = - A n r(-iy)

6.1.46

6.1.47

as z+m along any curve joining z=O and Z= m , providingz# --a, ---a-1, . . . ; zf --b, -b-1, . . . .

- (2n)! 1 2n r(n++)

1 ---

22n(n!)2-~ (n)=rtr(n+i)

- -&j [I-%+=*- * * *

1 1 1

(n+ OJ 1 Some Definite Integrals

6.1.50

e;‘~;~~’‘] T In r(z)=Jm[(z-l) e+- - ( 9 2 > 0)

=(z-+) In z-z++ In 21r

( 9 2 > 0) m arctan (t/z) dt

6.2. Beta Function

+2J, e”8-1

6.2.1

B(z,w)=J t s -1 ( 1 4 - 1 dt- -Jm&;*+. dt

= 2 r (sin t)s-1 (cos t ) tw-1 dt

( 9 2 > 0, aw > o >

r (z)r (w)=B(w,z) r(z+w)

6.3. Psi (Digamma) Function E-

#(z)=d[ln r (z)]/~z= r’(~)/r (z)

6.2.2 B(z,w)=

6.3.1

4 Some authors employ the special double factorial nota- tion as follows:

(24 ! 1 =2.4.6 . . . (24=2% i (h-1 ) ! I =1.3.5. . . ( 2 n - i ) = ~ 2” r(n++)

d 680meauthorswrite$(z)=~lnr(~+1) andeimilarlyfor

the polygamms functions.

FIGURE 6.2. Psi function.

y = $(z) = d In r kc)/&

Integer Values

n-1

k=1 6.3.2 #(l)=-’~, #(n)=-r+ Ck-’ (n22)

Fractional Values

6.3.3

#(+)=-7-2 In 2=--1.96351 00260 21423 . . .

6.3.4

#(n++)=-r-21n2+2 I+ ,+ . . - +:) 1 2n 1

(n 2 1)

( l

Recurrence Formulas

1 6.3.5

6.3.6

‘(n + ‘I= (n - 1) + z (n - 2) + z

t(Z+ l) = +(Z) + ;

l + 1 + . . .

1 1 +i&+,+,+9(1+4

GAMMA RTNCTION AND RELATED FUNCTIONS 259 Reflection Formula

6.3.7 +(l-z)=+(z)+* cot *z

Duplication Formula

6.3.8 +(22)=Mz)+++ (z+&) +In 2

Psi Function in the Complex Plane -

6.3.9 +GI =*(z> 6.3.10

9+(iy>=W+(-iy)=W+(l +iy)=W+(l -iy)

6-3-11 Y+(iy)=&/-'+#,~ coth xy

6.3.12 Y+(++iy) =&r tanh ?ry

1 63-13 j$(l+iy)=---+# ~ ~ 0 t h rv 2Y

= y g (n2+yS) -1 n-1

Series Expansions

6.3.14 +(l+~)=-r+C(-l)"~(n)~~-' (Iz1<1) n-2

6.3.15 +( 1 + 2) =& -1- &€ cot e- (1 -9) -'+ 1-7

- 5It(2n+ 1) - 11 ZSA (I z I <2) n-1

6.3.16

(~#-1,-2,-3, . . . )

6.3.17

9+(l+iy)=l-r-- 1 l+y2

+g (- l)"+'[r(2n+ 1) -l]y2' n=1

(IYl<2)

(- OJ <Y< -1

OD

= -r+ y2 c n-'(n*+yS) -1 a-1

Asymptotic Formulae

6.3.18

1 = Bz, -In z-s-n-l c- 2nz2"

=In z---- 1 1 1 1 + . . .

(z+- in lergzl<*)

22 1 2 9 + 1 2 0 2 4 - ~ 6

6.3.19

1 +-+. . . 1 1 =In y+-+- 12oy4 2 m Y 6

(Y+OJ)

Extremaoof r(z) - Zeros of $(z) 9 6

I

+l. 462 -0.504 -1.573 -2.611 -3.635 -4.653 -5.667 -6.678

+O. 886 -3.545 +2.302 -0.888 +O. 245 -0.053 +o. 009 -0.001

Zo=1.46163 21449 68362 r(xo)= .88560 31944 10889

6.3.20 zn=-n+(ln n)-'+o[(ln n)-*]

Definite Integrals

6.3.21

6From W. Sibagaki, Theory and applications of the gamma function, Iwanami Syoten, Tokyo, Japan, 1952 (with permission).

GAMMA FUNCTION AND RELATED FUNCTIONS 261 d

FIGURE 6.3. Incomplete gamma function. ?*(a,%)=- e-Lto-1dt

From F. G. Tricomi, Siilla funzione gamma incompleta, Annali di Matematica, IV, 33, 1950 (with permission). r(a) %-a r o

*See page n.

262 GAMMA FUNCTION AND RELATED F"CT1ONS

6.5.5 Probability Integral of the +Distribution

6.5.6

(Pearson's Form of the Incomplete Gamma Function)

m

6.5.7 C(z,a)=l tu-1 cos t dt (L@'a<l)

m

6.5.8 S(z,a)=$, ta-l sin t dt (9'a<l)

6.5.9 nm

6.5.11

Incomplete Gamma Function aa a Confluent Hypergeometric Function (eee chapter 13)

6.5.12 y(u,z)=a-lzue-tM(l, l+a,z)

=u-'zU M(a, l+a,-z)

Special Values

6.5.13

= 1 -e,,- (2) e-2

For relation to the Poisson distribution, 26.4.

6.5.14 r*(-n, z)=z"

6.5.15 I' (0, z)=le-'t-'dt=El (5)

see

6.5.16

6.5.17

6.5.18

6.5.19

6.5.20

6.5.21

6.5.22

6.5.23

6.5.24

Recurrence Formulas

9e-" P(a+l, z)=P(a, z)---- r(a+l>

y (a+ 1 ,z) = uy(a,z)

e-' V*(u-l,z) =m*(u,z) +- r (a)

Derivatives and Differential Equations

6.5.26

b" - ax" [x-T(u,s)~= (-i)nz-a-qa+n,z) (n=O, 1,2, . . .)

6.5.27 b" - bX" [e"z"~* (a,x)]=e"z"-"y*(a-n, z)

(n=O, 1,2, . . .)

Series Developmente

6.5.29

GAMMA FUNCTION AND RELATED FUNCTIOXS 263

Continued Fraction

6.5.31

Asymptotic Expansions

6.5.32

Suppose Rn(a,c")=un,,(a,z)+ . . . is the re- mnintlcr nftcr n terms in this series. Then if a , ~ nrc real, w e 11avr for n>a-2

!Iin(a,z)! I lun+,(a,z)l

ni i t l sign I?, (a,z) =sign u,<+, (a,z).

0 for a>1

1 for Osa<1

6.5.35

(z+m in I nrg zl<+r) Numeric

6.7. Use and Extension of the Tables

Example 1. Compute r(6.38) to 8s. Using the r~wirr~~i icc rchtioii 6.1.16 niitl Table 6.1 wc 1 1 avc,

r (6.38) = [ (5.38) (4.38) (3.38) (2.33) ( 1.38) ] r ( 1.38) = 232.4367 1.

Example 2. Compute In r(56.38), iisiiig Table 6.4 niid liiicnr iiitrrpolation iii j... \\-e liavc

Definite Integrals

6.5.36

6.6. Incomplete Beta Function

6.6.1 Br(a,b) =J2 t~-'(l--t)b-'d2 0

6.6.2 I r (a, b) = Br (ab) /B (a,b)

For statistical applications, see 26.5.

Symmetry

6.6.3 I,(a,b)=l --I,-r(b,u)

Helation to Binomial Expansion

For binomial distribution, see 26.1.

Recurrencc Formulas

6.6.5 Ir(U,b)=XIr(U- 1,b) + (l-~)IZ(a,b- 1)

6.6.6 (a+b-a)I,(a,b) =a(l-z)12(a+ 1,b- l>+bI,(a,b+ 1)

6.6.7 (~+b)l,(a,b) =al,(a+ 1,b) +bI,(a,b+ 1)

Relation to Hypergeometric Function

6.6.8 B,(a,b)=a-'~'c"F(a,l-b; a+l; Z)

:a1 Methods

*

The crror of liiicar intrrpolation in the table of tlic function f2 is smaller than lo-' in this region. Hence, f2(56.38) = .92041 67 and In I'(56.38) = 169.85497 42.

Direct interpolation in Table 6.4 of log,, r(n) climiiiatcs tlic necessity of employing logarithms. HOWCVP~, tlic rrror of liiicar intcrpolation is .002 so tltnt log,, r(n) is obtained with a rclativc error of 10-5.

*See page 11. In r(56.38) = (56.38-3) In (56.38) - (56.38)

+ j 2 (56.38)

264 GAMMA FUNCTION AN

Example 3. Compute $(6.38) to 8s. Using the recurrence relation 6.3.6 and Table 6.1.

=1.77275 59.

Example 4. Compute (L(56.38). Using Table 6.3 we have $(56.38)=ln 56.38-j3(56.38).

The error of linear interpolation in the table of the function f3 is smaller than 8XlO-' in this region. Hence,f3(56.38)=.00889 53and$(56.38)= 4.023219.

Example 5. Compute In I'(1-i). From the reflection principle 6.1.23 and Table 6.7, In r(1-i) =In r(l+i) = -.6509+.3016i.

Example 6. Compute In F(+++i). Taking the logarithm of the recurrence relation 6.1.15 we have,

In r (&++i) =In r (#++i) -In (*+&i) - - -.23419+.03467i

-(& In *+i arctan 1) = .11239- .75073i

The logarithms of complex numbers are found from 4.1.2.

Example 7. Compute In I'(3+7i) using the duplication formula 6.1.18. Taking the logarithm of 6.1.18, we have

-4 In 2r=- .91894 (#+7i) In 2= 1.73287+ 4.852036 In r(#+$i)=-3.31598+ 2.32553i In r(2+4$=-2.66047+ 2.938693

In r(3+7i) =-5.16252+10.11625i

Example 8. Compute In I'(3+7i) to 5D using the asymptotic formula 6.1.41. We have

In (34-79 =2.03022 15+1.16590 45i.

Then,

(2.5+7i) In (3+7i)=-3. 0857779+17.1263119i - (3+7i) = -3.0000000- 7. oooooooi 4 In (2~)= .9189385

[12(3+7i)]-'= .00431037 .01005753 -[360(3+7i)3]-i= . 0000059- . 0000022i

In r(3+7i)=-5. 16252 +io. 11625i

RELATED FUNCTIONS

6.8. Summation of Rational Series by Means of Polygamma Functions

An infinite series whose general term is a rational function of the index may always be reduced to a finite series of psi and polygamma functions. The method will be illustrated by writing the ex- plicit formula when the denominator contains a triple root.

Let the general term of an infinite series have the form

where p(n) is a polynomial of degree m + 2r + 3s - 2 at most and where the constants a,, pi., and yf are distinct. Expand un in partial fractions as follows

OD

Then, we may express u, in terms of the

constants appearing in this partial fraction expansion as follows

n-1

Higher order repetitions in the denominator are If the denominator contains handled similarly.

GAMMA FUNCTION AND RELATED FUNCTIONS 265

only simple or double roots, omit the correapond- ing lines.

Example 9. Find - 1

Since

we have

a1=1, a2=3, as=*, al=*, &=-l, *=#.

Thus,

8= -)$(2) +$(13) -#$(It) =.047198.

Example 10.

m 1

we have,

Therefore

S= 16~(1)-16$(1~) +$'(1) +$'(li) =.013499.

Example 11.

(see also 6.3.13). 1 m

Evaluate 8 = c n-l (n2+ 1) (n*+4)

We have, 1

i -i -i i Hence, al=-, 6 e=-, 6 &=- 12 ' a 4 = 3

a1=i, az=-i, aa=2i, a,=-2i,

and therefore

s=- --z [$( 1 +i) -$(1 -ill +a i [$(1+2i) -$(1-2i)l. 6

By 6.3.9, this reduces to

1 1 3 6 8=- Y$(l+i)-- 9$(1+2{).

From Table 6.8, s=.13876.

References Texts

[6.1] E. Artin, Einfiihrung in die Theorie der Gamms- funktion (Leipzig, Germany, 1931).

[6.2] P. E. Bohmer, Differenzengleichungen und be- stimmte Integrale, chs. 3, 4, 5 (K. F. Koehler, Leipzig, Germany, 1939).

16.31 G. Doetsch, Handbuch der Laplace-Transforma- tion, vol. 11, pp. 52-61 (Birkhauser, Basel, Switzerland, 1955).

[6.4] A. Erdblyi et al., Higher transcendental functions, vol. 1, ch. 1, ch. 2, sec. 5; vol. 2, ch. 9 (McGraw- Hill Book Co., Inc., New York, N.Y., 1953).

[6.5] C. Hastings, Jr., Approximations for digital computers (Princeton Univ. Press, Princeton, N.J., 1955).

[6.6] F. Losch and F. Schoblik, Die Fakultiit und ver- wandte Funktionen (B. G. Teuhner, Leipzig, Germany, 1951).

[6.7] W. Sibagaki, Theory and applications of the gamma function (Iwanami Syoten, Tokyo, Japan, 1952).

[6.S] E. T. Whittaker and G. N. Watson, A course of modern analysis, ch. 12, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952).

Tables

[6.9] A. Abramov, Tables of I n r(z) for complex argu- Translated from the Russian by D. G. ment.

Fry (Pergamon Press, New York, N.Y., 1960). In r(z+iy), z=O(.Ol)lO, y=0(.01)4, 6D.

[6.10] Ballistic Research Laboratory, A table of the factorial numbers and their reciprocals from l! through lOOO! to 20 significant digits. Technical Note NO. 381, Aberdeen Proving Ground, Md., 1951.

[6.11] British Association for the Advancement of Science, Mathematical tables, vol. 1, 3d ed., pp. 40-59 (Cambridge Univ. Press, Cambridge, England, 1951). The gamma and polygamma functions.

Also l+l ' log lD (t)!dt, z=O(.Ol)l, 10D.

[6.12] H. T. Davis, Tables of the higher mathematical functions, 2 vols. (Principia Press, Bloomington, Ind., 1933, 1935). Extensive, many place tables of the gamma and polygamma functions up to $(4)(z) and of their logarithms.

[6.13] F. J. Duarte, Nouvelles tables de log,, nl 8,33 d6ci- males depuis n= l jusqu'h n=3000 (Kundig, Geneva, Switzerland; Index Generalis, Paris, France, 1927).


[6.14] National Bureau of Standards, Tables of nl and r(n+& for the first thousand values of n, Ap- plied Math. Series 16 (U.S. Government Printing O5ce, Washington, D.C., 1951). nf, 16S;r(n+&, 8s.

[6.15] National Bureau of Standards, Table of Coulomb wave functions, vol. I, pp. 114-135, Applied Math. Series 17 (U.S. Government Printing O5ce, Washington, D.C., 1952).

9 [ryi + is)/r (1 + is] ,9 = o(.oo5) 2 (.oi) 6 (.02)1 o(. 1 20 (.2) 60( .5) 1 10,l OD ; apg r (1 + is) ,s = O(.el) 1 (.02)

3 (.05)10(.2)20(.4)30(.5)85, 8D.

[6.16] National Bureau of Standards, Table of the gamma function for complex arguments, Applied Math. Series 34 (U.S. Government Printing O5ce, Washington, D.C., 1954).

In r(z+iy), z=d(.l)lO, y=0(.1)10, 12D.

Contains an extensive bibliography. (6.171 National Physical Laboratory, Tables of Weber

parabolic cylinder functions, pp. 226-233 (Her Majesty’s Stationery Office, London, England, 1955).

Real and imaginary parts of In r(ik+$ia), k-0(1)3, a = 0 (. 1) 5(. 2) 20, 8D ; (IF (4 + +ia) /r (+ + tis) 1) -I”

~=0(.02)1(.1)5(.2)20, 8D.

[6.18] E. S. Pearson, Table of the logarithms of the com- plete r-function, arguments 2 to 1200, Tracts for Computers No. VI11 (Cambridge Univ. Press, Cambridge, England, 1922). Loglo r(p), p=2(.1) 5(.2)70(1)1200, 10D.

[6.19] J. Peters, Ten-place logarithm tables, vol. I, Ap- pendix, pp. 58-68 (Frederick Ungar Publ. Co., New York, N.Y., 1957). nl, n=1(1)60, exact;

18D. (n!)-’, n=1(1)43, 54D; Log,o(nl), n=1(1)1200,

(6.20) J. P. Stanley and M. V. Wilkes, Table of the reciprocal of the gamma function for complex argu- ment (Univ. of Toronto Press, Toronto, Canada, 1950). Z= -.5( .01).5, y=O(.Ol)l, 6D.

I6.211 M. Zycakowski, Tablice funkcyi eulera i pokrewnych (Panstwowe Wydawnictwo Naukowe, Warsaw, Poland, 1954). Extensive tables of integrals involving gamma and beta functions.

For references to tabular material on the incomplete gamma and incomplete beta functions, see the references in chapter 26.

chapter 6 gamma function and related functions

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polygamma

gamma function

gamma

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