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In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
, Mittag-Leffler's theorem concerns the existence of
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are ''poles'' of the function. ...
s 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 In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its Zero of a function, zeroes. The theorem m ...
, which asserts existence of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s with prescribed zeros. The theorem is named after the Swedish mathematician
Gösta Mittag-Leffler Magnus Gustaf "Gösta" Mittag-Leffler (16 March 1846 – 7 July 1927) was a Sweden, Swedish mathematician. His mathematical contributions are connected chiefly with the theory of functions that today is called complex analysis. He founded the pre ...
who published versions of the theorem in 1876 and 1884.


Theorem

Let U be an
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
in \mathbb C and E \subset U be a subset whose
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x contains a point of S other than x itself. A ...
s, 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_^ \frac. Then there exists a meromorphic function f on U whose
poles Pole or poles may refer to: People *Poles (people), another term for Polish people, from the country of Poland * Pole (surname), including a list of people with the name * Pole (musician) (Stefan Betke, born 1967), German electronic music artist ...
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 In complex analysis, a removable singularity of a holomorphic function is a point at which the function is Undefined (mathematics), undefined, but it is possible to redefine the function at that point in such a way that the resulting function is ...
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 In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deri ...
function on U.


Proof sketch

One possible proof outline is as follows. If E is finite, it suffices to take f(z) = \sum_ p_a(z). If E is not finite, consider the finite sum S_F(z) = \sum_ 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 and E = \mathbb^+, 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_^\infty \frac . This series converges normally on any
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
subset of \mathbb \smallsetminus \mathbb^+ (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: \pi\csc(\pi z) = \sum_ \frac = \frac1z + 2z\sum_^\infty \frac \pi\sec(\pi z) = \sum_ \frac = 2\sum_^\infty \frac \pi\cot(\pi z) = \lim_\sum_^N \frac1 = \frac1z + 2z\sum_^\infty \frac1 \pi\tan(\pi z) = \lim_\sum_^N \frac = 2z\sum_^\infty \frac (\pi\csc(\pi z))^2 = \sum_ \frac1 (\pi\sec(\pi z))^2 = \sum_ \frac1


See also

*
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It re ...
* Liouville's theorem * Mittag-Leffler condition of an inverse limit * Mittag-Leffler summation * Mittag-Leffler function


References

*. *.


External links

* {{springer, title=Mittag-Leffler theorem, id=p/m064170 Theorems in complex analysis