Question

I think this is a trick question, but I am not sure where the "trick" is:

I am told to evaluate the integral\[\oint_\gamma {1\over 1+w^2}dw\]where \(w\in \mathbb C\) and \(\gamma\) is (i) a circular path centered the origin or (ii) an elliptical path centered the origin.

The thing is I believe that since the integrand is holomorphic in the enclosed regions and on the boundaries, Cauchy's theorem should imply that they are both zero? But I think they are unlikely to give 2 questions with a trivial answer, so I am likely to be mistaken somewhere.

Thank you.

Answer 1

The only question is whether or not the path includes the poles of the function
\[ f(w) = \frac{1}{1+w^2} \]
If they do not, then on that path and in its interior, the function is holomorphic and the integral is zero.

If they do include one or both of the simple poles \( w = \pm i \), then the integral is equal to
\[ orientation(\gamma) . 2\pi \sum_{w \ \in \ poles} Res[f,w] \]

Answer 2

*\( 2 \pi i \)