(where n is a function of k.)
(Cauchy's formula)

Sameed
I think your proof needs to make it clear which contour the is referring to in each case. So, for example, in the first case I think you should write:
to indicate the closed contour which encloses all the poles (as stated in the exercise), and then:
(where n is a function of k.)
with an explanation of what the are (maybe with a diagram?).

Also an explanation of how travelling round the for all is equivalent to travelling round would be very useful to other members of the forum when trying to understand your proof.

Oh! Just spotted a typo here -- very tiny, but the contour by the integral sign is actually a lower-case "r", when it should be an upper-case gamma .

But thanks for the solution -- I couldn't quite think of the right way to start this one.