Number Theory/Gamma function

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The Gamma function is a result of the efforts to expand the factorials onto the complex plane.



Definition

In order to fulfill the necessary realation/condition:



we can set the Gamma function to be:



whereby , because by partial integration we get:



If and we receive for , when .


Theorem: poles of the function

The poles of the function appear at , whereby ; the residues are at and its contour integral is equal to .

Proof

The Euler definition of Legendre for the function is:



Now, we can split the integral, which results in:



However, for we can substitute its series representation:



in the first integral, which leads to:



Since we have a polynomial in the first integral, we can use the power rule and we get the following:



If we examine the denominator of the series, we can see, that for:



poles occur, with which we can obtain the residue:



If we use now the residue theorem, we can evalute the following contour integral:



As the series:


is equal to the sum



with , which is , the integral is equal to: