Mittag-Leffler's Theorem

If a function analytic at the origin has no singularities other than poles for finite , and if we can choose a sequence of contours about tending to infinity such that never exceeds a given quantity on any of these contours and is uniformly bounded on them, then

where is the sum of the principal parts of at all poles within . If there is a pole at , then we can replace by the negative powers and the constant term in the Laurent series of about .