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.

Portrait of Gösta Mittag-Leffler

The theorem is named after the Swedish mathematician Gösta Mittag-Leffler who published versions of the theorem in 1876 and 1884.^[1][2][3]

Theorem

Let ${\displaystyle U}$ be an open set in ${\displaystyle \mathbb {C} }$ and ${\displaystyle E\subset U}$ be a subset whose limit points, if any, occur on the boundary of ${\displaystyle U}$. For each ${\displaystyle a}$ in ${\displaystyle E}$, let ${\displaystyle p_{a}(z)}$ be a polynomial in ${\displaystyle 1/(z-a)}$ without constant coefficient, i.e. of the form $${\displaystyle p_{a}(z)=\sum _{n=1}^{N_{a}}{\frac {c_{a,n}}{(z-a)^{n}}}.}$$ Then there exists a meromorphic function ${\displaystyle f}$ on ${\displaystyle U}$ whose poles are precisely the elements of ${\displaystyle E}$ and such that for each such pole ${\displaystyle a\in E}$, 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 ${\displaystyle p_{a}(z)}$. Furthermore, any other meromorphic function ${\displaystyle g}$ on ${\displaystyle U}$ with these properties can be obtained as ${\displaystyle g=f+h}$, where ${\displaystyle h}$ is an arbitrary holomorphic function on ${\displaystyle U}$.

Proof sketch

One possible proof outline is as follows. If ${\displaystyle E}$ is finite, it suffices to take ${\textstyle f(z)=\sum _{a\in E}p_{a}(z)}$. If ${\displaystyle E}$ is not finite, consider the finite sum ${\textstyle S_{F}(z)=\sum _{a\in F}p_{a}(z)}$ 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 ${\displaystyle U}$ (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 $${\displaystyle p_{k}(z)={\frac {1}{z-k}}}$$ and ${\displaystyle E=\mathbb {Z} ^{+}}$, Mittag-Leffler's theorem asserts the existence of a meromorphic function ${\displaystyle f}$ with principal part ${\displaystyle p_{k}(z)}$ at ${\displaystyle z=k}$ for each positive integer ${\displaystyle k}$. More constructively we can let $${\displaystyle f(z)=z\sum _{k=1}^{\infty }{\frac {1}{k(z-k)}}.}$$

This series converges normally on any compact subset of ${\displaystyle \mathbb {C} \smallsetminus \mathbb {Z} ^{+}}$ (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: $${\displaystyle \pi \csc(\pi z)=\sum _{n\in \mathbb {Z} }{\frac {(-1)^{n}}{z-n}}={\frac {1}{z}}+2z\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{z^{2}-n^{2}}}}$$ $${\displaystyle \pi \sec(\pi z)=\sum _{n\in \mathbb {Z} }{\frac {(-1)^{n-1}}{z-\left(n+{\frac {1}{2}}\right)}}=2\sum _{n=0}^{\infty }{\frac {(-1)^{n-1}(n+{\frac {1}{2}})}{z^{2}-(n+{\frac {1}{2}})^{2}}}}$$ $${\displaystyle \pi \cot(\pi z)=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{z-n}}={\frac {1}{z}}+2z\sum _{n=1}^{\infty }{\frac {1}{z^{2}-n^{2}}}}$$ $${\displaystyle \pi \tan(\pi z)=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {-1}{z-(n+{\frac {1}{2}})}}=2z\sum _{n=0}^{\infty }{\frac {-1}{z^{2}-(n+{\frac {1}{2}})^{2}}}}$$ $${\displaystyle (\pi \csc(\pi z))^{2}=\sum _{n\in \mathbb {Z} }{\frac {1}{(z-n)^{2}}}}$$ $${\displaystyle (\pi \sec(\pi z))^{2}=\sum _{n\in \mathbb {Z} }{\frac {1}{(z-(n+{\frac {1}{2}}))^{2}}}}$$

See also

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