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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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