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.

Theorem

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

Proof sketch

One possible proof outline is as follows. If $E$ is finite, it suffices to take ${\textstyle f(z)=\sum _{a\in E}p_{a}(z)}$. If $E$ is not finite, consider the finite sum ${\textstyle S_{F}(z)=\sum _{a\in F}p_{a}(z)}$ where $F$ is a finite subset of $E$. While the $S_{F}(z)$ may not converge as F approaches E, one may subtract well-chosen rational functions with poles outside of $U$ (provided by Runge's theorem) without changing the principal parts of the $S_{F}(z)$ 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 $$p_{k}(z)={\frac {1}{z-k}}$$ and $E=\mathbb {Z} ^{+}$, Mittag-Leffler's theorem asserts the existence of a meromorphic function $f$ with principal part $p_{k}(z)$ at $z=k$ for each positive integer $k$. More constructively we can let $$f(z)=z\sum _{k=1}^{\infty }{\frac {1}{k(z-k)}}.$$

This series converges normally on any compact subset of $\mathbb {C} \smallsetminus \mathbb {Z} ^{+}$ (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: $$\tan(z)=\sum _{n=0}^{\infty }{\frac {8z}{(2n+1)^{2}\pi ^{2}-4z^{2}}}$$ $$\csc(z)=\sum _{n\in \mathbb {Z} }{\frac {(-1)^{n}}{z-n\pi }}={\frac {1}{z}}+2z\sum _{n=1}^{\infty }(-1)^{n}{\frac {1}{z^{2}-(n\,\pi )^{2}}}$$ $$\sec(z)\equiv -\csc \left(z-{\frac {\pi }{2}}\right)=\sum _{n\in \mathbb {Z} }{\frac {(-1)^{n-1}}{z-\left(n+{\frac {1}{2}}\right)\pi }}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n+1)\pi }{(n+{\frac {1}{2}})^{2}\pi ^{2}-z^{2}}}$$ $$\cot(z)\equiv {\frac {\cos(z)}{\sin(z)}}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{z-n\pi }}={\frac {1}{z}}+2z\sum _{k=1}^{\infty }{\frac {1}{z^{2}-(k\,\pi )^{2}}}$$ $$\csc ^{2}(z)=\sum _{n\in \mathbb {Z} }{\frac {1}{(z-n\,\pi )^{2}}}$$ $$\sec ^{2}(z)={\frac {d}{dz}}\tan(z)=\sum _{n=0}^{\infty }{\frac {8((2n+1)^{2}\pi ^{2}+4z^{2})}{((2n+1)^{2}\pi ^{2}-4z^{2})^{2}}}$$ $${\frac {1}{z\sin(z)}}={\frac {1}{z^{2}}}+\sum _{n\neq 0}{\frac {(-1)^{n}}{\pi n(z-\pi n)}}={\frac {1}{z^{2}}}+\sum _{n=1}^{\infty }{(-1)^{n}}{\frac {2}{z^{2}-(n\,\pi )^{2}}}$$

See also

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

References

• 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]