Using Cauchy Integral Formula
I thought I was a good math student, but for some reason I am getting tripped up on complex analysis. Can someone please explain what I am doing wrong when I try to solve the following problem?
Problem: Given the function , where , compute the "smart" way - using the Cauchy Integral Formula.
If a=0, the problem is trivial, so assume a not zero. Cauchy's integral formula gives us that if is any "nice" curve enclosing 0 in the complex plane, then
Let's take to be a counter-clockwise circle about 0, of small enough radius that the singularity is excluded from the interior (so that Cauchy's integral formula applies. Now, we want to compute the integral
Immediately using the parametrization doesn't appear to be fruitful here, but the substitution yields:
where is now a circle of radius . Hence, this curve contains a singularity, . Where do I go from here? Any better ideas? (My book says to use Cauchy integral formula, so I'd appreciate solutions which somehow use that result.)