Mittag-Leffler theorem

The Mittag-Leffler theorem is a fundamental result in complex analysis that helps us reconstruct a complex function based on its behavior at certain points. Specifically, if a function has specified singularities (points where it behaves badly) at certain locations, the theorem guarantees we can build an earlier part of the function—called a "meromorphic function"—that has those same singularities and principal parts, while being as simple as possible elsewhere. Essentially, it allows us to piece together a function with prescribed local behaviors, providing a way to understand and recreate complex functions from their singularities.