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 Adrien-Marie 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 well-known value of the Gamma function at a non-integer 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 non-positive integers:
-
Here γ is the
Euler-Mascheroni constant.
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