Gamma function

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This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

The Gamma funciton is defined as

Relations and values

Factorial

Using integration by parts,

At , goes to 0. Using L'Hopital's rule, it's easy to show that .

So,

Thus,

Using this, and the the fact that , then we can get the factorial function. For a positive integer,

z=1/2

For z=1/2,

Substituting ,

where in the last line we used the fact that is an even function. The integral is called the Gaussian integral and has a well known value of .

Thus,