Mittag-Leffler's Theorem
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Mittag-Leffler's 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.

## Theorem

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 ${\displaystyle p_{a}(z)}$.

One possible proof outline is as follows. If ${\displaystyle E}$ is finite, it suffices to take ${\displaystyle f(z)=\sum _{a\in E}p_{a}(z)}$. If ${\displaystyle E}$ is not finite, consider the finite sum ${\displaystyle S_{F}(z)=\sum _{a\in F}p_{a}(z)}$ where ${\displaystyle F}$ is a finite subset of ${\displaystyle E}$. While the ${\displaystyle S_{F}(z)}$ may not converge as F approaches E, one may subtract well-chosen rational functions with poles outside of D (provided by Runge's theorem) without changing the principal parts of the ${\displaystyle S_{F}(z)}$ and in such a way that convergence is guaranteed.

## Example

Suppose that we desire a meromorphic function with simple poles of residue 1 at all positive integers. With notation as above, letting

${\displaystyle p_{k}={\frac {1}{z-k}}}$

and ${\displaystyle E=\mathbb {Z} ^{+}}$, Mittag-Leffler's theorem asserts (non-constructively) the existence of a meromorphic function ${\displaystyle f}$ with principal part ${\displaystyle p_{k}(z)}$ at ${\displaystyle z=k}$ for each positive integer ${\displaystyle k}$. This ${\displaystyle f}$ has the desired properties. More constructively we can let

${\displaystyle f(z)=z\sum _{k=1}^{\infty }{\frac {1}{k(z-k)}}}$.

This series converges normally on ${\displaystyle \mathbb {C} }$ (as can be shown using the M-test) to a meromorphic function with the desired properties.

## Pole expansions of meromorphic functions

Here are some examples of pole expansions of meromorphic functions:

${\displaystyle \tan(z)=\sum \limits _{n=0}^{\infty }{\dfrac {8z}{(2n+1)^{2}\pi ^{2}-4z^{2}}}}$
${\displaystyle \csc(z)=\sum _{n\in \mathbb {Z} }{\frac {(-1)^{n}}{z-n\pi }}={\frac {1}{z}}+2z\sum _{n=1}^{\infty }(-1)^{n}{\frac {1}{z^{2}-(n\,\pi )^{2}}}}$
${\displaystyle \sec(z)\equiv -\csc \left(z-{\frac {\pi }{2}}\right)=\sum _{n\in \mathbb {Z} }{\frac {(-1)^{n-1}}{z-\left(n+{\frac {1}{2}}\right)\pi }}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n+1)\pi }{(n+{\frac {1}{2}})^{2}\pi ^{2}-z^{2}}}}$
${\displaystyle \cot(z)\equiv {\frac {\cos(z)}{\sin(z)}}=\sum _{n\in \mathbb {Z} }{\frac {1}{z-n\pi }}={\frac {1}{z}}+2z\sum _{k=1}^{\infty }{\frac {1}{z^{2}-(k\,\pi )^{2}}}}$
${\displaystyle \csc ^{2}(z)=\sum _{n\in \mathbb {Z} }{\frac {1}{(z-n\,\pi )^{2}}}}$
${\displaystyle \sec ^{2}(z)={\dfrac {d}{dz}}\tan(z)=\sum \limits _{n=0}^{\infty }{\dfrac {8((2n+1)^{2}\pi ^{2}+4z^{2})}{((2n+1)^{2}\pi ^{2}-4z^{2})^{2}}}}$
${\displaystyle {\frac {1}{z\sin(z)}}={\frac {1}{z^{2}}}+\sum _{n\neq 0}{\frac {(-1)^{n}}{\pi n(z-\pi n)}}={\frac {1}{z^{2}}}+\sum _{n=1}^{\infty }{(-1)^{n}}{\frac {2}{z^{2}-(n\,\pi )^{2}}}}$

## References

• Ahlfors, Lars (1953), Complex analysis (3rd ed.), McGraw Hill (published 1979), ISBN 0-07-000657-1.
• Conway, John B. (1978), Functions of One Complex Variable I (2nd ed.), Springer-Verlag, ISBN 0-387-90328-3.