From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Newsgroups: sci.math
Subject: Re: Alternate definition of Gamma function
Date: 24 Jul 1998 19:46:40 -0400

In article <6paot4$9gj$1@winter.news.erols.com>, TTL <tlim@erols.com> wrote:
:If my calculation is correct, the following can be used as a definition
:of Gamma function for all complex numbers s not = 0,-1,-2....:
 
: i (2 sin (s*pi))^(-1)  Int_{infty}^{infty} (-x)^(s-1) e^(-x) dx
:
:where Int_{infty}^{infty} represents a contour integral going from
:infinity to 0 along "upper" real line (i.e. arg(-z)= - pi ) to C, a
:circle centered at 0 oriented counterclockwise, going around C back to
:the real line and continuing along the "lower" real line to infinity.
:
:Does anyone see any books that define Gamma function in this way,
:as an extension of real Gamma function?
:

 I did not check all details of your representation but it is closely
related to two formulas due to Hankel (1863), and they are related by the
complement formula Gamma(s)*Gamma(1-s) = pi/sin(pi*s). 

Hankel as found in 
Special Functions, by Nico M. Temme, John Wiley 1996,
ISBN 0-471-11313-1

1/Gamma(z) = (1/(2*pi*i)) int_L(s^(-z)*exp(s)) ds

where L runs from -inf (arg(s)=-pi) around 0 counterclockwise and back to
-inf (arg(s)=pi)

Hankel as found in the Encyclopaedic Dictionary of Mathematics
The MIT Press, Cambridge 1980
ISBN 0-262-59010-7

(except for a small typo) 

Gamma(z) = (1/(exp(2*pi*i*z) - 1))*int_C(t^(z-1)*exp(-t)) dt

where C runs from +infinity around 0 counterclockwise and back to
+infinity.

Your representation (if correct) can probably be obtained by a
simple change of variable.

:Most books that I read use infinite series.
:
 You mean infinite products? (Or, infinite series in Prym's decomposition,
also found in Temme's book?)

Hope it helps, ZVK(Slavek).