ABSTRACT
Generalized Gamma Function
1.1 The Gamma Function ()
The problem of giving x! a useful meaning when x is any nonnegative number
was solved by Euler (1707{1783) in his private correspondence of October 13,
1729 and January 8, 1730 with Goldbach (1690{1764), who dened what is
now called the gamma function
() :=
Z
t
e
t
dt ( > 0; = + i): (1.1)
The notation () and the name gamma function were introduced by Legen-
dre (1752{1833). The representation (1.1) may be thought of as representing
the application of x
+
to the test function '(x) = e
x
for 0 x < 1 as
follows:
() := hx
+
; e
x
i :=
Z
x
e
x
dx ( > 0; = + i); (1.2)
where '(x) = e
x
is known to be in space S, the space of innitely dieren-
tiable functions which, together with their derivatives, approaches zero more
rapidly than any power of 1=jxj as jxj ! 1. Using the general regularization
relation ([118], p. 48),
hx
+
; 'i :=
Z
x
"
'(x)
n
X
k=0
x
k
k!
