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:

${\displaystyle \tan(z)=\sum \limits _{n=0}^{\infty }{\dfrac {8z}{(2n+1)^{2}\pi ^{2}-4z^{2}}}}$

${\displaystyle \csc(z)=\sum _{n\in \mathbb {Z} }{\frac {(-1)^{n}}{z-n\pi }}={\frac {1}{z}}+2z\sum _{n=1}^{\infty }(-1)^{n}{\frac {1}{z^{2}-(n\,\pi )^{2}}}}$

${\displaystyle \sec(z)\equiv -\csc \left(z-{\frac {\pi }{2}}\right)=\sum _{n\in \mathbb {Z} }{\frac {(-1)^{n-1}}{z-\left(n+{\frac {1}{2}}\right)\pi }}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n+1)\pi }{(n+{\frac {1}{2}})^{2}\pi ^{2}-z^{2}}}}$

${\displaystyle \cot(z)\equiv {\frac {\cos(z)}{\sin(z)}}=\sum _{n\in \mathbb {Z} }{\frac {1}{z-n\pi }}={\frac {1}{z}}+2z\sum _{k=1}^{\infty }{\frac {1}{z^{2}-(k\,\pi )^{2}}}}$

${\displaystyle \csc ^{2}(z)=\sum _{n\in \mathbb {Z} }{\frac {1}{(z-n\,\pi )^{2}}}}$

${\displaystyle \sec ^{2}(z)={\dfrac {d}{dz}}\tan(z)=\sum \limits _{n=0}^{\infty }{\dfrac {8((2n+1)^{2}\pi ^{2}+4z^{2})}{((2n+1)^{2}\pi ^{2}-4z^{2})^{2}}}}$

${\displaystyle {\frac {1}{z\sin(z)}}={\frac {1}{z^{2}}}+\sum _{n\neq 0}{\frac {(-1)^{n}}{\pi n(z-\pi n)}}={\frac {1}{z^{2}}}+\sum _{n=1}^{\infty }{(-1)^{n}}{\frac {2}{z^{2}-(n\,\pi )^{2}}}}$

See also

• Riemann-Roch theorem
• Liouville's theorem
• Mittag-Leffler condition of an inverse limit
• Mittag-Leffler summation
• Mittag-Leffler function

References

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (September 2015) (Learn how and when to remove this template message)

• 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.

External links

• "Mittag-Leffler theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Mittag-Leffler's theorem". PlanetMath.