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The Gamma funciton is defined as
Relations and values
Using integration by parts,
At t = 0, tz − 1( − e − t) goes to 0. Using L'Hopital's rule, it's easy to show that .
- = (z − 1)Γ(z − 1).
- Γ(z) = (z − 1)Γ(z − 1).
Using this, and the the fact that Γ(1) = 1, then we can get the factorial function. For n a positive integer,
- Γ(n) = (n − 1)!.
Substituting x = t1 / 2,
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 .