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 $a \in E$, f(z) − p_a(z) is holomorphic at a. In particular, the principal part of f at a is p_a(z).

One possible method of proof 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. Use Runge's theorem to construct a rational function R_F(z) approximating S_F(z). As E is exhausted by the finite sets F, the approximations R_F(z) will approach a limit, which is the desired meromorphic function f.

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 = 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.

See also

• Riemann-Roch theorem
• Weierstrass factorization theorem

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's theorem on PlanetMath