§ 13.2
67
BASIC MATHEMATICAL FORMULAS
The infinite products
~- = r(z)
fr (1 + Z_-=--l)e-(Z-1)/n = zeYz n(1 + ~)e-z!n n n
eY(z-l)
n~l
n~l
(2)
define the function for all complex values of z. y = 0.577216
r(z) =
lim n~CX)
(3)
(1 . 2 . 3 .'. n)n z
-c--~'c-
z(z + 1) (z
+ 2) ... (z + n + 1)
(4)
13.2. Functional equations
r(z + 1) =
zr(z),
r(z)r(l -
r( =) re ~ 1) ... r( z + ~ -1)n z r( ~) r(z;
1)2 z- 1
=
TT
z)
=~.~.
!2
=
1
SlllTTZ
(2TT)(n-l)!~r(z)
y';r(z)
(1) (2)
(3)
13.3. Special values
r(l)=O!=l,
r(n)=(n-1)!=1·2·3 ... (n-1)
for n a positive integer.
r(~)=y';
(1) (2)
13.4. Logarithmic l1erivative
tjJ(z)
=
dIn - - r(z) = - y + I'D dz n=l
• dtjJ(z) tjJ (z) = ~dZ
~
=
(1n
1)
~--~-
z
+n-
1
1
f:'o (z + n)2
(1)
(2)
If the terms of a convergent series are rational functions of n, by a partial fraction decomposition the series may be summed in terms of tjJ and its derivatives by the series of this section.
13.5. Asymptotic expressions. If 0 means" of the order of," In r(x)
=
In [V2TT XX-l!2e- X]
+ _1_ + O(~) 12x x3
(1)