How can we calculate this generalized integral:

Thanks in advance!!

Printable View

- July 9th 2009, 10:28 AMypatiaHelp with a Generalized integral
How can we calculate this generalized integral:

Thanks in advance!! - July 9th 2009, 11:12 AMRandom Variable
Let

There are two simple poles in the upper half complex plane

and

and since is an even function

- July 9th 2009, 11:15 AMgalactus
There are many ways to skin an integral, but here goes:

The integral is given by

We can find the residues by considering the complex function:

The singularities are at

The equation can be solved by .

But, remember that

The 4 roots are:

Two of the roots are in the upper half plane and two in the lower half

plane. Don't pay attention to the roots in the lower half plane because we

are choosing a closed semicircle in the upper half plane as our contour.

We consider the singularities that are inside the contour, so the others do not contribute anything.

Now, lets compute the residues:

This whittles down to:

So, the residues corresponding to the pole at is given by

Therefore, hence, heretofore, and whence the integral evaluates to:

Now, because we want from 0 to infinity, divide by 2 and we get:

We could also do this by regular old integration by trying a partial fraction:

This leads to the rather onerous looking:

For the first one, the sub will simplify it nicely.

For the second one, try letting - July 9th 2009, 08:14 PMsimplependulum

Sub.

The integral =

In general ,