This integral is convergent for n > 0.

Since integrals of this type occur so frequently, it becomes convenient to develop and tabulate several key relationships for future use. In particular, three such expressions associated with the gamma function are given below.

Gamma Function Relationships

The remainder of this subsection formally develops these three relationships and gives a simple application of their use.

Proof that, for any positive n, .

To see this, we have from eqn. (8.10) that

Now integrating by parts, with , we let

Therefore,

Proof that, for a positive integer, .

If n is a positive integer, then

or, in general, [where is sometimes referred to as the generalized factorial function].

Proof that, for n = 1/2, .

Setting n = 1/2 in the basic definition gives,

Letting gives and putting this result into the integral reduces the original expression to

Squaring this result gives

Now switching to polar coordinates with and , we have

and with the u,v domain limits defining the first quadrant, , the limits on r and become .

Therefore, the above expression becomes

Thus, we have shown that .

An Example

As a simple example of the use of the gamma function, consider the following integral,

Letting , this becomes

Thus, with the use of the gamma function, evaluating this integral is quite straightforward.