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.


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 .


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: