I've been asked to find the number of distinct values that $ \oint_\gamma \frac{dz}{(z-a_1)(z-a_2)...(z-a_n)}$ can take for simple closed curves $ \gamma $ not passing through any of the $ a_i $.
My thoughts so far:
I know I should be thinking about partial fractions and using Cauchy's Integral Formula. WLOG set $ |a_1| < |a_2| < ... < |a_n| $. Let $ I(\gamma) $ denote the interior of the closed curve $ \gamma $. Now there are $ n $ possible scenarios that could yield different values for the integral: $ a_1 \in I(\gamma) $, $ a_1 $ and $ a_2 \in I(\gamma) $, ... , $ a_1, a_2, ... , a_n \in I(\gamma) $. From the integral formula, each root $ a_i $ of $ p(z) = (z-a_1)...(z-a_n) $ will make a non-zero contribution iff $ a_i \in I(\gamma) $. Writing as partial fractions, we have:
$ \frac{1}{p(z)} = \frac{A_1}{z-a_1} + \frac{A_2}{z - a_2} + ... + \frac{A_n}{z - a_n} $, which leads to
$ A_1 (z-a_2)...(z-a_n) + A_2(z-a_1)...(z -a_n) + ... + A_n(z-a_1)...(z-a_{n-1}) = 1$ (*)
And so we see that the possible values of the integral are $ 2\pi i \left( A_1 \right) $, or $ 2\pi i \left( A_1 + A_2 \right) $, or ... $ 2 \pi i \left( A_1 + ... + A_n \right) $.
But the relationship (*) gives restrictions on the values of the $ A_i $. For example, $ A_1 + ... + A_n = 0 $ (by considering the coefficient of $ z^{n+1} $).
I'm unsure about making the final leap to the answer - how should I count the possible values of the $n $ sums? I would greatly appreciate any advice, either in the form of pointing out mistakes in my reasoning so far or hints on where to go next. I'd rather not have a full answer (or a hint that makes my task trivial). Thanks.