Gamma function
In mathematics, the Gamma function is a function that extends the concept of factorial to the complex numbers. The notation was thought of by AdrienMarie Legendre. If the real part of the complex number z is positive, then the integral

converges absolutely. Using
integration by parts, one can show that
Because of Γ(1) = 1, this relation implies

for all
natural numbers n. It can further be used to extend Γ(
z) to a
holomorphic function defined for all complex numbers
z except
z = 0, − 1,− 2, − 3, ... by
analytic continuation.
It is this extended version that is commonly referred to as the Gamma function.
The Gamma function does not have any zeros.
Perhaps the most wellknown value of the Gamma function at a noninteger is

The Gamma function has a
pole of order 1 at
z = −
n for every
natural number n; the
residue there is given by
The following multiplicative form of the Gamma function is valid for all complex numbers
z which are not nonpositive integers:

Here γ is the
EulerMascheroni constant.