Contour Integration for Evaluating Difficult Real Integrals

[fred_91]

Homework Statement

I want to evaluate the following integral:

\int \frac{\sqrt{a^2-x^2}(a-x)^{-1/c}(x-d)}{x^2-b^2}dx

Homework Equations

The Attempt at a Solution

I rewrote it in the following form (to try to make it simpler):

\int \frac{\sqrt{a+x}(a-x)^{1/2-1/c}(x-d)}{x^2-b^2}dx

I have no idea how to start integrating this. I have tried putting it into Mathematica, but Mathematica didn't evaluate it.
Any ideas will be very much appreciated.

 

[christoff]

We know some things about the integrand;
It has singularities at \pm b.
If c is positive, there is a singularity at x=a.
It is complex for x>a.

If this were a definite integral, then I would suggest some kind of contour integration approach. But actually finding an antiderivative for that? Good luck, friend.

 

[fred_91]

If this integral was a definite integral, with the integration limits being:
d (for the lower limit), and f (for the upper limit);

would this be able to integrate using contour integration? If so, do you have any hints how I can go through that approach?

Thank you.

 

[christoff]

Possibly. Contour integration methods for computing real integrals are really a bit of an art. There is no general procedure; the way you do it depends on what your real integral is. Also, in your case, it also depends on what your lower limit d, and upper limit f, are with respect to the singularities mentioned.

If you haven't learned any complex analysis (eg. Cauchy-Goursat theorem, Residue Theorem, Cauchy integral formula, etc.), then there really is no way I can guide you to attempt to solve this problem in only a few lines. So I will first recommend you read the following wikipedia article, to get a general sense of how the procedure works.

http://en.wikipedia.org/wiki/Methods_of_contour_integration#Applications_of_integral_theorems

Then, if you feel up to it, take out any decent textbook on the subject of Complex Analysis from your library, and it should have a section on the evaluation of Real integrals by contour integration.

I should warn you that I personally have never tried working out an integral as messy as the one you have. In theory, you should be able to do it, I just don't have the time or drive to try. Complex analysis is very useful, and the theoretical results are beautiful, but the applications (like this) are tedious, and at times very difficult.

Good luck.