13.1 The Gamma Function

The gamma function, denoted as ?( n), is also known as generalized factorial function. It is defined as

and this improper ^[*] integral converges (approaches a limit) for all n > 0.

We will derive the basic properties of the gamma function and its relation to the well known factorial function

We will evaluate the integral of (13.1) by performing integration by parts using the relation

Letting

we get

Then, with (13.3), we write (13.1) as

With the condition that n > 0, the first term on the right side of (13.6) vanishes at the lower limit, that is, for x = 0. It also vanishes at the upper limit as x ? ?. This can be proved with L' H pital's rule ^[*] by differentiating both numerator and denominator m times, where m ? n. Then,

Therefore, (13.6) reduces to

and with (13.1) we have

By comparing the two integrals of (13.9), we see that

or

It is convenient to use (13.10) for n < 0, and (13.11) for n > 0.

From (13.10), we see that ?( n) becomes infinite as n ? 0.

For n =