### Gamma function

The gamma function and gamma distribution are two different things. This post is about the former and is a preparatory step to study the latter.

**Definition**. The **gamma function** is defined by

The integrand is smooth on so its integrability is determined by its behavior at and . Because of the exponent, it is integrable in the neighborhood of The singularity at is integrable if In all calculations involving the gamma function one should remember that its argument should be positive.

## Properties

1) Factorial-like property. Integration by parts shows that

if

2) because

3) Combining the first two properties we see that for a natural

Thus the gamma function extends the factorial to non-integer

4)

Indeed, using the density of the standard normal we see that

(replacing )

(replacing )

Many other properties are not required in this course.

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