Mittag-Leffler's theorem

In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. 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 $f(z)-p_a(z)$ is holomorphic 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}}+\sum _{{n=1}}^{\infty }(-1)^{n}{\frac {2z}{z^{2}-n^{2}\pi ^{2}}}

\cot(z)\equiv {\frac {\cos(z)}{\sin(z)}}=\sum _{{n\in {\mathbb {Z}}}}{\frac {1}{z-n\pi }}={\frac {1}{z}}+\sum _{{k=1}}^{\infty }{\frac {2z}{z^{2}-k^{2}\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}-\pi ^{2}n^{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), "Mittag-Leffler theorem", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
"Mittag-Leffler's theorem". PlanetMath.

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