Also apologies if this thread is in the wrong section. I wasn't sure whether to put this under calculus or something else...
I'm having difficulty in using contour Integration to evaluate the integral:
Firstly I found that the function being integrated is an even function so this is equivalent to finding:
Also where Re dentoes the Real part of exp(ix) function. The integral in question is now:
so the complex function in question is:
which is holomorphic everywhere in the complex plane except for the points ai, and -ai. That is what I've found so far.
What contour do I have to use? Also how would you integrate it because I'm seeing there will be much difficulty integrating with the z being replaced by the suitable parameterization.
The part on the x axis (this is what you are looking for) and
the part on the circle above (or below) the x axis. You will want to show that this is equal to zero then
If the 2nd integral goes to zero and the first integral is over the x-axis you have found the value you are looking for.
See this example on wiki
Residue theorem - Wikipedia, the free encyclopedia