Mittag-Leffler theorem

In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. Conversely, it can be used to express any meromorphic function as a sum of partial fractions. It is sister to the Weierstrass factorization theorem, which asserts existence of holomorphic functions with prescribed zeros. It is named after Gösta Mittag-Leffler.

Let ${\displaystyle D}$ be an open set in ${\displaystyle \mathbb {C} }$ and ${\displaystyle E\subset D}$ a closed discrete subset. For each ${\displaystyle a}$ in ${\displaystyle E}$, let ${\displaystyle p_{a}(z)}$ be a polynomial in ${\displaystyle 1/(z-a)}$. There is a meromorphic function ${\displaystyle f}$ on ${\displaystyle D}$ such that for each ${\displaystyle a\in E}$, the function ${\displaystyle f(z)-p_{a}(z)}$ has only a removable singularity at ${\displaystyle a}$. In particular, the principal part of ${\displaystyle f}$ at ${\displaystyle a}$ is ${\displaystyle p_{a}(z)}$.

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