[math-fun] Partial fraction decomposition for the reciprocal of an infinite product

I was recently reminded that the reciprocal of the famous infinite product sin(pi*z) / pi = z * Product_{1 <= n < oo} (1 - z^2 / n^2) can be expressed as pi / sin(pi*z) = Sum_{-oo < n < oo} (-1)^n / (z - n) (where the right side denotes the limit of the symmetrical finite sums). Cool, right? So I'm curious if there are any general rules for passing from an infinite product to the partial fraction decomposition of its reciprocal, or vice versa. Do people know other nice examples independent of the one above? —Dan