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

One possible proof outline is as follows. Notice that if $E$ is finite, it suffices to take $f(z) = \sum_{a \in E} p_a(z)$ . If $E$ is not finite, consider the finite sum $S_F(z) = \sum_{a \in F} p_a(z)$ 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 D (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}={\frac {1}{z-k}}

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

f(z) = z\sum_{k=1}^\infty \frac{1}{k(z-k)}

.

This series converges normally on $\mathbb{C}$ (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:

{\frac {1}{\sin(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}}}

\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}}}

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

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Mittag-Leffler theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
"Mittag-Leffler's theorem". PlanetMath.

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