6. Gamma Function and Related Functions

6. Gamma Function and Related Functions
PHILIPJ. DAVIS
Contents
Page
Mathematical Properties. . . . .
6.1.
6.2.
6.3.
6.4.
6.5.
6.6.
.
Gamma Function. . . . . .
Beta Function . . . . . . .
Psi (Digamma) Function. . .
Polygamma Functions. . . .
Incomplete Gamma Function.
Incomplete Beta Function. .
Numerical Methods . . . . . . .
.
. . . . . . . . . . . . . .
255
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255
258
258
260
260
263
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. . . . . .
263
.
.
.
.
.
.
.
.
.
.
6.7. Use and Extension of the Tables. . . . . . . . . . . . .
6.8. Summation of Rational Series by Means of Polygamma Functions. . . . . . . . . . . . . . . . . . . . . . . . .
263
264
. . . . . . . . . . . .
265
Table 6.1. Gamma, Digamma and Trigamma Functions (1 5 s l 2 ) . .
267
References.
. . . .
.
r(x),~n r(x),
,
. . . . . . .
,
+(z),+'(z), ~=i(.oo5)2, IOD
Table 6.2. Tetragamma and Pentagamma Functions (1 5 x 5 2 ) . . .
+"(x),
$J3'(2),~=1(.01)2,
1OD
Table 6.3. Gamma and Digamma Functions for Integer and HalfInteger Values ( l l n 5 1 0 1 ) . . . . . . . . . . . . . . . . . .
r(n), 11s
l/r(n), 9s
r(n+$), 8s
27 1
272
+(n), IOD
n!/[(2?r)h"+3]~",8D
[email protected]),
8D
n=1(1.)101
Table 6.4. Logarithms of the Gamma Function (1 I
loglo
r(n>, 8s
log10 r(n+#), 8 s
n 5 101). .
. . . .
274
log10 r(n+$), 8s
In r(n)-(n-$) lnn+n, 8D
log10 r(n+$), 8s
n= l(1) 101
National Bureau of Standards.
253
254
GAMMA FUNCTION AND RELATED FUNCTIONS
Table 6.5. Auxiliary Functions for Gamma and Digamma Func. .
Table 6.6. Factorials for Large Arguments ( 1 0 0 5 n S 1000) . . . .
Page
276
.
276
Table 6.7. Gamma Function for Complex Arguments. . . . . . . .
277
n!, n= 100(100) 1000, 20s
In r(z+iy), 2=1(.1)2, y=0(.1)10,
12D
Table 6.8. Digamma Function for Complex Arguments . . .
+(z+iy),2=1(.1)2, y=0(.1)10,
.
. .
.
288
5D
%‘+U+iy),10D
B’+(l+iy)-ln y, y l = . l l (-.Ol)O,
8D
The author acknowledges the assistance of Mary Orr in the preparation and checking of
the tables; and the assistance of Patricia Farrant in checking the formulas.
=.57721 56649. . .
Y is known as Euler's constant and is given to 25
decimal places in chapter 1. r(z) is single valued
and analytic over the entire complex plane, save
for the points z=-n(n=O, 1, 2, . . . ) where it
possesses simple poles with residue (- 1) "/n!. Its
reciprocal i/r(z) is an entire function possessing
simple zeros at the points z= -n(n=O, 1, 2, . . .).
-=-s
Hankel's Contour Integral
6.1.4
1
(-t)-'e-'dt
i
r(z) 2,
(k<l
c
FIGURE6.1. Gumma function.
, y-r(z),
n(z)=z!=r(z+i)
6.1.10
r (n+ 3) =1.5.9.13.4". . (4n-3) r(t>
r(+)=3.6256099082. . .
1.4-7.10. . . (3n-2)
6.1.11 r(n+#)=
3"
6.1.12
r(n+1)=1.2.3 . . . (n-l)n=n!
6.1.7
lim -=o=
1
z+,
r(-z)
1
(-n-l)!
r(n+$) =
I-()) = 2 s me-12dt=&=1.77245 38509 . . . =(-3)!
0
1-3-5-7... (2n-1)
2"
2.5.8-11.. . (3n-1)
6.1.13 l"(n+#)=
3"
r (3)
r(3)
r(#)=i.3aii 79394. . .
(n=O, 1, 2, . . .)
Fractional Values
6.1.8
r (4)
r($)=2.67893 85347 . . .
Integer Values
6.1.6
Y=l/r(4
r(3/2)=$,*=.8~622692%. . . =(3)!
Factorial and II Notations
6.1.5
- - -,
6.1.9
=)
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.
-
*
6.1.14
r(n+i)= 3.7.11.15.4". . (4n-1) Ut)
r($)=i.22541 67024 .
*See page 11.
. .
255
256
GAMMA FUNCTION AND RELATED FUNCTIONS
Recurrence Formulas
6.1.15
6.1.29
r(i~)r(-iy)=ir(i~)iz=9r
y sinh 7ry
r(z+i)=Zr(Z)=Z!=Z(Z-i)!
6.1.16
r(n+~)=(n-il+z)(n-2+~) . . . (i+z)r(i+z)
= (n- 1+ z)!
6.1.31
r (1 +iy)r (1-iy)= Ir (1 +iY) (L“
?/
sinh ry
. . (l+z)z!
=(n-l+z)(n-2+2).
Reflection Formula
6.1.17
r(z>r(i-z)=-zr(-z)r(z)=t
=J=,
a c 7rz
tz-1
0
-dt
l+t
(O<
9 2
<
1)
Power Series
6.1.33
In r (1+ z)= -In (1 +z) +z (1 -7)
+5( - ~ ) ~ ~ ~ ( ~ (14<2>
~ - ~ l ~ ” / ~
Duplication Formula
n-2
6.1.18
r(2~)=(2~):+
222-3
r(z)r(z++)
Triplication Formula
{(n)is the Riemann Zeta Function (see chapter
23).
Series Expansion * for 1 /r(2)
6.1.19 r(3z)= ( 2 ~-1) 35’4r (2)r (z+#r(z++)
6.1.34
Gauss’Multiplication Formula
k
Binomial Coefficient
Pochhammer’sSymbol
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
6.1.24
[email protected])=r);
In r(Z)=In r(z)
arg r(z+l)=arg r(z)+arctan
X
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
ck
1.00000
0.57721
-0.65587
-0.04200
0. 16653
-0.04219
-0.00962
0.00721
-0.00116
-0.00021
0.00012
-0.00002
-0.00000
0.00000
-0.00000
0.00000
0.00000
-0.00000
0.00000
0.00000
-0.00000
0.00000
-0.00000
-0.00000
0.00000
0.00000
00000
56649
80715
26350
86113
77345
19715
89432
51675
52416
80502
01348
12504
11330
02056
00061
00050
00011
00001
00000
00000
00000
00000
00000
00000
00000
000000
015329
202538
340952
822915
555443
278770
466630
918591
741149
823882
547807
934821
272320
338417
160950
020075
812746
043427
077823
036968
005100
000206
000054
000014
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.
257
GAMMA FUNCTION AND RELATED F"CX'I0NS
Error Term for Asymptotic Expansion
Polynomial Approximations'
6.1.35
6.1.42
01x51
r (x+ 1) =z! = 1+-alx+ -a& +62+ag4+-a&+
E(Z)
55x10-6
57486 46
.95123 63
=-.69985 88
~l=-.
~ 4 =
Uz=
U6=-.
~ 3
If
R,(z)= In r (z)-(z-#
In z+z-+ In ( 2 ~ )
,42455 49
10106 78
-5
,,12rn(2rn--l)z
B2m
an--l
then
where
bl=-. 57719
bs= .98820
b3= -. 89705
bq=
,91820
1652
5891
6937
6857
75670
be= .48219
b7= -.
19352
bs= .03586
bs=-.
4078
9394
7818
8343
Stirling's Formula
K(z)=upperU l bound(z2/(u2+z3)
O
I
For z real and positive, R, is less in absolute value
than the first term neglected and has the same
ign.
6.1.43
6.1.37
9th r(iy)=%?ln r(-iy)
-& In (2.) -w-+ln y,
(y++
6.1.44
A n r(iy)=arg r(iy)=-arg
= - A n r(-iy)
r(-iy)
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-+)
+z
m
In z-z++
B2m
2rn(2m-1)2*-'
I n
(2~)
(z+m
in larg
z~<T)
For B, see chapter 23
6.1.46
6.1.47
6.1.41
In
r(z)-(z-&
1
In z-z++ In (%)+---1
122 360z3
From C. Hmtings, Jr., Approximations for digital
computers, Princeton Univ. Press, Princeton, N.J., 1955
(with permission).
as z+m along any curve joining z=O and Z=
providingz# --a, ---a-1, . . . ; zf --b, -b-1,
. . . .
m,
(2n)!
---
1 2n
r(n++)
22n(n!)2-~ (n)=rtr(n+i)
1
1
1
-&j [I-%+=*-
-
* * *
1
FIGURE
6.2. Psi function.
y = $(z) = d In r kc)/&
(n+ OJ 1
Some Definite Integrals
6.1.50
Integer Values
In r(z)=Jm[(z-l)
T
(92
> 0)
( 9 2
> 0)
e+- e;‘~;~~’‘]-
=(z-+) In z-z++ In 21r
m arctan (t/z)dt
+2J,
e”8-1
6.2. Beta Function
ts-1
(14-1
dt--Jm&;*+.
6.3.3
dt
6.3.4
#(n++)=-r-21n2+2
6.2.2
) :+
1
2n 1
l
aw>o>
(n 2 1)
B(z,w)= r (z)r (w)=B(w,z)
r(z+w)
#(z)=d[ln r (z)]/~z=
r’(~)/r
(z)
4 Some authors employ the special double factorial notation as follows:
( 2 4 ! 1 =2.4.6 . . . ( 2 4 = 2 % i
( h - 1 ) ! I =1.3.5. . . ( 2 n - i ) = ~ 2”r(n++)
d
680meauthorswrite$(z)=~lnr(~+1)
andeimilarlyfor
the polygamms functions.
I+,+..(
6.3. Psi (Digamma) Function E6.3.1
Ck-’ (n22)
k=1
Fractional Values
= 2 r (sin t)s-1 (cos t ) t w - 1 dt
( 9 2 > 0,
n-1
#(l)=-’~,
#(n)=-r+
#(+)=-7-2 In 2=--1.96351 00260 21423 . . .
6.2.1
B(z,w)=J
6.3.2
Recurrence Formulas
6.3.5
t(Z+
l) = +(Z)
1
+;
6.3.6
l
+
1
‘(n+‘I= (n -1) +z (n-2) +z
+...
1
1
+i&+,+,+9(1+4
259
GAMMA RTNCTION AND RELATED FUNCTIONS
6.3.19
Reflection Formula
+(l-z)=+(z)+* cot *z
6.3.7
Duplication Formula
1
=In y+-+-
+(22)=Mz)+++(z+&) +In 2
6.3.8
Psi Function in the Complex Plane
-
+GI=*(z>
6.3.9
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
63-13
1
j$(l+iy)=---+#
2Y
=y
~ ~ 0 rv
t h
n-1
Series Expansions
6.3.14
+(l+~)=-r+C(-l)"~(n)~~-'
(Iz1<1)
n-2
&€
cot e-(1-9) -'+ 1-7
6.3.20
-n-1
5It(2n+ 1) -11ZSA
-
Zeros of $(z)
I
+ l . 462
-0.504
-1.573
-2.611
-3.635
-4.653
-5.667
-6.678
+O. 886
-3.545
+2.302
-0.888
+O. 245
-0.053
+o. 009
-0.001
Zo=1.46163 21449 68362
r(xo)=.88560 31944 10889
6.3.15
+( 1 +2) =&-1-
9
6
g (n2+yS) -1
..
(Y+OJ)
Extremaoof r(z)
6.3.10
1
+-+.1
12oy4 2 m Y 6
(Iz I <2)
zn=-n+(ln n)-'+o[(ln n)-*]
Definite Integrals
6.3.21
6.3.16
(~#-1,-2,-3,
...)
6.3.17
9+(l+iy)=l-r--
1
l+y2
+g(- l)"+'[r(2n+1) -l]y2'
n=1
= -r+ y2
c n-'(n*+yS)
(IYl<2)
OD
-1
a-1
(-
Y<
OJ
-1
Asymptotic Formulae
6.3.18
1
-In z-s-n-l
=In z----1
22
= Bz,
c-
2nz2"
1
1
1
1 2 9 + 1 2 0 2 4 - ~ 6+ . . .
(z+- in lergzl<*)
6From W. Sibagaki, Theory and applications of the
gamma function, Iwanami Syoten, Tokyo, Japan, 1952
(with permission).
GAMMA FUNCTION AND RELATED FUNCTIONS
261
d
FIGURE6.3. Incomplete gamma function.
?*(a,%)=-
r
r(a)
%-a
o
e-Lto-1dt
From F. G. Tricomi, Siilla funzione gamma incompleta, Annali di Matematica, IV, 33, 1950 (with permission).
*See page n.
GAMMA FUNCTION AND RELATED FUNCTIONS
261
d
FIGURE6.3. Incomplete gamma function.
?*(a,%)=-
r
r(a)
%-a
o
e-Lto-1dt
From F. G. Tricomi, Siilla funzione gamma incompleta, Annali di Matematica, IV, 33, 1950 (with permission).
*See page n.
262
GAMMA FUNCTION AND RELATED F"CT1ONS
6.5.5
6.5.16
Probability Integral of the +Distribution
6.5.17
6.5.18
6.5.6
(Pearson's Form of the Incomplete Gamma Function)
6.5.19
6.5.20
Recurrence Formulas
m
6.5.7
C(z,a)=l tu-1 cos t dt
([email protected]'a<l)
9e-"
P(a+l, z)=P(a, z)---r(a+l>
6.5.21
m
6.5.8
S(z,a)=$,
ta-l
sin t dt
(9'a<l)
6.5.9
nm
6.5.22
y (a+1,z)=uy(a,z)
6.5.23
V*(u-l,z)
e-'
=m*(u,z)+-r (a)
Derivatives and Differential Equations
6.5.24
6.5.11
Incomplete Gamma Function aa a Confluent
Hypergeometric Function (eee chapter 13)
6.5.12
6.5.26
b"
y(u,z)=a-lzue-tM(l, l+a,z)
=u-'zU
ax" [x-T(u,s)~= (-i)nz-a-qa+n,z)
(n=O, 1,2, . . .)
M(a, l+a,-z)
6.5.27
b"
Special Values
bX" [e"z"~*(a,x)]=e"z"-"y*(a-n, z)
6.5.13
(n=O, 1,2,. . .)
=1-e,,-
(2)e-2
For relation to the Poisson distribution, see
26.4.
6.5.14
6.5.15
r*(-n, z)=z"
I' (0, z)=le-'t-'dt=El
(5)
Series Developmente
6.5.29
263
GAMMA FUNCTION AND RELATED FUNCTIOXS
Definite Integrals
6.5.36
*
Continued Fraction
6.5.31
6.6. Incomplete Beta Function
Asymptotic Expansions
6.6.1
Br(a,b)=J2
6.6.2
I r (a,b)
6.5.32
0
t~-'(l--t)b-'d2
= Br (ab)/B(a,b)
For statistical applications, see 26.5.
Symmetry
Suppose Rn(a,c")=un,,(a,z)+ . . . is the remnintlcr nftcr n terms in this series. Then if a , ~
nrc real, w e 11avr for n>a-2
!Iin(a,z)!I
niitl
lun+,(a,z)l
I,(a,b)=l --I,-r(b,u)
6.6.3
Helation to Binomial Expansion
For binomial distribution, see 26.1.
sign I?,(a,z) =sign u,<+,(a,z).
Recurrencc Formulas
0 for a>1
6.6.5
Ir(U,b)=XIr(U-
6.6.6
(a+b-a)I,(a,b)
1,b) + (l-~)IZ(a,b- 1)
=a(l-z)12(a+ 1,b- l>+bI,(a,b+ 1)
1 for Osa<1
6.5.35
6.6.7
(~+b)l,(a,b)=al,(a+ 1,b)+bI,(a,b+ 1)
Relation to Hypergeometric Function
(z+m in I nrg +
<)zlr
6.6.8
B,(a,b)=a-'~'c"F(a,l-b; a + l ; Z)
Numeric:a1 Methods
6.7. Use and Extension of the Tables
Example 1. Compute r(6.38) to 8s. Using
cc
6.1.16 niitl Table 6.1 wc
the r ~ w i r r ~ ~ i irchtioii
1 1 avc,
r (6.38)= [(5.38)(4.38)(3.38)(2.33)(1.38)]r (1.38)
= 232.43671.
Example 2. Compute In r(56.38), iisiiig Table
6.4 niid liiicnr iiitrrpolation iii
\\-e liavc
j...
In r(56.38) = (56.38-3) In (56.38)- (56.38)
+j 2 (56.38)
The crror of liiicar intrrpolation in the table of
tlic function f 2 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.
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.
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 explicit formula when the denominator contains a
triple root.
Let the general term of an infinite series have
the form
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 *+iarctan 1)
= .11239- .75073i
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
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
OD
Then, we may express
n-1
u, in terms of the
constants appearing in this partial fraction expansion as follows
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) = -3.00000007. 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
Higher order repetitions in the denominator are
handled similarly. If the denominator contains
265
GAMMA FUNCTION AND RELATED FUNCTIONS
only simple or double roots, omit the correaponding lines.
Therefore
S=
+$'(li)
=.013499.
16~(1)-16$(1~)+$'(1)
Example 9. Find
-
Example 11.
1
1
(see also 6.3.13).
n-l (n2+1) (n*+4)
m
Evaluate 8
= c
Since
1
We have,
we have
a1=1,
a2=3,
as=*, al=*, &=-l, *=#.
i e=-,
-i6
Hence, al=-,6
Thus,
8=
-)$(2)
a1=i,
+$(13) -#$(It) =.047198.
i
az=-i, aa=2i, a,=-2i,
and therefore
Example 10.
m
-i
&=-12 ' a 4 = 3
s=- --z
1
[$(1
+i)-$(1 -ill
6
+ai [$(1+2i) -$(1-2i)l.
By 6.3.9, this reduces to
1
Y$(l+i)--61 9$(1+2{).
3
8=-
we have,
From Table 6.8, s=.13876.
References
Texts
Tables
[6.1] E. Artin, Einfiihrung in die Theorie der Gammsfunktion (Leipzig, Germany, 1931).
[6.9] A. Abramov, Tables of
r(z) for complex argument. Translated from the Russian by D. G.
Fry (Pergamon Press, New York, N.Y., 1960).
In r(z+iy), z=O(.Ol)lO, y=0(.01)4, 6D.
[6.2] P. E. Bohmer, Differenzengleichungen und bestimmte Integrale, chs. 3, 4, 5 (K. F. Koehler,
Leipzig, Germany, 1939).
16.31 G. Doetsch, Handbuch der Laplace-Transformation, 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 (McGrawHill 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 verwandte 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).
I n
[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 ' l o g l D (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
$(4)(z)and of their logarithms.
[6.13] F. J. Duarte, Nouvelles tables de log,, nl 8,33 d6cimales depuis n = l jusqu'h n=3000 (Kundig,
Geneva, Switzerland; Index Generalis, Paris,
France, 1927).
266
GAMMA FUNCTION AND RELATED FUNCTIONS
[6.14] National Bureau of Standards, Tables of nl and
r(n+& for the first thousand values of n, Applied Math. Series 16 (U.S.Government Printing
O5ce, Washington, D.C., 1951). nf, 16S;r(n+&,
8s.
[6.15] National Bureau of Standards, Table of Coulomb
wave functions, vol. I, pp. 114-135, Applied
Math. Series 17 (U.S. Government Printing
O5ce, Washington, D.C., 1952).
9[ryi+is)/r(1 +is],9 =o(.oo5)2 (.oi)6 (.02)1o(.1
20 (.2)60(.5)1 10,lOD; apg r (1+is),s = O(.el)1 (.02)
3 (.05)10(.2)20(.4)30(.5)85, 8D.
[6.16] National Bureau of Standards, Table of the gamma
function for complex arguments, Applied Math.
Series 34 (U.S. Government Printing O5ce,
Washington, D.C., 1954).
In r(z+iy), z=d(.l)lO, y=0(.1)10, 12D.
Contains an extensive bibliography.
(6.171 National Physical Laboratory, Tables of Weber
parabolic cylinder functions, pp. 226-233 (Her
Majesty’s Stationery Office, London, England,
1955).
Real and imaginary parts of In r(ik+$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 complete 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
pendix, pp. 58-68
New York, N.Y.,
(n!)-’, n=1(1)43,
18D.
logarithm tables, vol. I, Ap(Frederick Ungar Publ. Co.,
1957). nl, n=1(1)60, exact;
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 argument (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.