Is there a generalization of Cauchy's integral formula in case the integrand has more than 1 pole?

Question

Cauchy's integral formula reads

$$ \frac{1}{2\pi i} \oint \frac{dz}{(z-z_0)}f(z) = f(z_0)$$

So, my question is what is the results for the following integral

$$ \frac{1}{2\pi i} \oint \frac{dz}{\prod_i(z-z_i)}f(z) $$, if it exists of course? Is it just the sum of the contours over each pole separately?

Answer 1

Here is what you are looking for $$\frac{1}{2\pi i} \oint \frac{dz}{\prod_i(z-z_i)}=\frac{1}{2\pi i}\sum_i \oint\frac{\alpha_i dz}{(z-z_i)} $$ where $$\frac{1}{\prod_i(z-z_i)} = \sum_i \frac{\alpha_i dz}{(z-z_i)}$$

for example $$\frac{1}{(z-1)(z-2)}=\frac{1}{(z-2)}-\frac{1}{(z-1)} $$