Gamma and Beta functions

Definition.
For x positive we define the Gamma function by

This integral cannot be easily evaluated in general, therefore we first look at the Gamma function at two important points. We start with x = 1:

Now we look at the value at x = 1/2:

The last integral cannot be evaluated using antiderivative (see the note on Newton integrability). However, this particular definite integral is very important (for instance in probability), so people eventually found a trick to find its value.

To find the value of the Gamma function at other points we deduce an interesting identity using integration by parts:

The limit is evaluated using l'Hospital's rule several times. We see that for x positive we have

If we apply this to a positive integer n, we get

So we see that the Gamma function is a generalization of the factorial function. It is possible to show that the limit of the Gamma function at 0 from the right is infinity, the graph looks like this:

Since at integer points, the value of the Gamma function is given by the factorial, it follows that the Gamma function grows to infinity even faster than exponentials.

Definition.
For x,y positive we define the Beta function by

Using the substitution u = 1 - t it is easy to see that

To evaluate the Beta function we usually use the Gamma function. To find their relationship, one has to do a rather complicated calculation involving change of variables (from rectangular into tricky polar) in a double integral. This is beyond the scope of this section, but we include the calculation for the sake of completeness:

Thus


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