Mittag-Leffler's theorem explained

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.

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

be an open set in and be a subset whose limit points, if any, occur on the boundary of . For each in , let be a polynomial in without constant coefficient, i.e. of the form$$p\_a(z)\; =\; \backslash sum\_^\; \backslash frac.$$Then there exists a meromorphic function on whose poles are precisely the elements of and such that for each such pole , the function has only a removable singularity at ; in particular, the principal part of at is . Furthermore, any other meromorphic function on with these properties can be obtained as , where is an arbitrary holomorphic function on .

Proof sketch

One possible proof outline is as follows. If

is finite, it suffices to take $f(z)\; =\; \backslash sum\_\; p\_a(z)$. If is not finite, consider the finite sum $S\_F(z)\; =\; \backslash sum\_\; p\_a(z)$ where is a finite subset of . While the may not converge as F approaches E, one may subtract well-chosen rational functions with poles outside of (provided by Runge's theorem) without changing the principal parts of the 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 $$p\_k(z)\; =\; \backslash frac$$ and

, Mittag-Leffler's theorem asserts the existence of a meromorphic function with principal part at for each positive integer . More constructively we can let $$f(z)\; =\; z\backslash sum\_^\backslash infty\; \backslash frac\; .$$

This series converges normally on any compact subset of

(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:$$\backslash tan(z)\; =\; \backslash sum\_^\backslash infty\; \backslash frac$$$$\backslash csc(z)=\; \backslash sum\_\; \backslash frac=\; \backslash frac\; +\; 2z\backslash sum\_^\backslash infty\; (-1)^n\; \backslash frac$$$$\backslash sec(z)\backslash equiv\; -\backslash csc\backslash left(z-\backslash frac\backslash right)=\; \backslash sum\_\; \backslash frac=\; \backslash sum\_^\backslash infty\; \backslash frac$$$$\backslash cot(z)\backslash equiv\; \backslash frac=\; \backslash lim\_\backslash sum\_^N\; \backslash frac=\; \backslash frac\; +\; 2z\backslash sum\_^\backslash infty\; \backslash frac$$$$\backslash csc^2(z)\; =\; \backslash sum\_\; \backslash frac$$$$\backslash sec^2(z)\; =\; \backslash frac\backslash tan(z)=\; \backslash sum\_^\backslash infty\; \backslash frac$$$$\backslash frac=\; \backslash frac\; +\; \backslash sum\_\; \backslash frac=\; \backslash frac\; +\; \backslash sum\_^\backslash infty\; \backslash frac$$

See also

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

References

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Notes and References

1. Mittag-Leffler . 1876 . En metod att analytiskt framställa en funktion af rational karakter, hvilken blir oändlig alltid och endast uti vissa föreskrifna oändlighetspunkter, hvilkas konstanter äro på förhand angifna . Öfversigt af Kongliga Vetenskaps-Akademiens förhandlingar Stockholm . 33 . 6 . 3–16.
2. Mittag-Leffler . 1884 . Sur la représentation analytique des fonctions monogènes uniformes dʼune variable indépendante . Acta Mathematica . 4 . 1–79. 10.1007/BF02418410 . 124051413 . free .
3. Turner . Laura E. . 2013-02-01 . The Mittag-Leffler Theorem: The origin, evolution, and reception of a mathematical result, 1876–1884 . Historia Mathematica . en . 40 . 1 . 36–83 . 10.1016/j.hm.2012.10.002 . 0315-0860. free .