Mittag-Leffler's theorem

Mittag-Leffler's theorem prescribes the principal parts of a meromorphic function. Given an open subset $D$ of the complex plane, a subset $A$ having no accumulation point in $D$, and a finite sum of negative powers $P_a(z)={\sum }_{j=1}^{m_a}{c}_{a,j}(z-a{)}^{-j}$ for each $a\in A$, there is a meromorphic function on $D$ with exactly these principal parts and no other poles. The difference of any two such meromorphic functions is a holomorphic function on $D$, because their principal parts cancel at every prescribed pole.