|
|
| Line 1: |
Line 1: |
| In [[complex analysis]], '''Mittag-Leffler's theorem''' concerns the existence of [[meromorphic function]]s with prescribed [[pole (complex analysis)|pole]]s. It is sister to the [[Weierstrass factorization theorem]], which asserts existence of [[holomorphic function]]s with prescribed [[Zero (complex analysis)|zeros]]. It is named after [[Gösta Mittag-Leffler]].
| | The individual who wrote the post is called Jayson Hirano and he completely digs that title. phone psychic readings ([http://www.january-yjm.com/xe/index.php?mid=video&document_srl=158289 just click the next post]) For years she's been operating as a journey agent. My wife and I live in Mississippi but now I'm contemplating other options. As a woman what she really likes is fashion and she's been performing it for quite a while.<br><br>My web site [http://medialab.zendesk.com/entries/54181460-Will-You-Often-End-Up-Bored-Try-One-Of-These-Hobby-Ideas- free tarot readings] - free psychic readings ([http://myfusionprofits.com/groups/find-out-about-self-improvement-tips-here/ myfusionprofits.com]) |
| | |
| ==Theorem==
| |
| Let <math>D</math> be an [[open set]] in <math>\mathbb C</math> and <math>E \subset D</math> a [[closed set|closed]] [[discrete set|discrete]] subset. For each <math>a</math> in <math> E</math>, let <math>p_a(z)</math> be a polynomial in <math>1/(z-a)</math>. There is a meromorphic function <math>f</math> on <math>D</math> such that for each <math>a \in E</math>, <math>f(z)-p_a(z)</math> is holomorphic at <math>a</math>. In particular, the principal part of <math>f</math> at <math>a</math> is <math>p_a(z)</math>.
| |
| | |
| One possible proof outline is as follows. Notice that if <math> E </math> is finite, it suffices to take <math> f(z) = \sum_{a \in E} p_a(z)</math>. If <math>E</math> is not finite, consider the finite sum <math> S_F(z) = \sum_{a \in F} p_a(z)</math> where <math> F </math> is a finite subset of <math>E</math>. While the <math>S_F(z)</math> 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 <math>S_F(z)</math> and in such a way that convergence is guaranteed.
| |
| | |
| ==Example==
| |
| | |
| Suppose that we desire a meromorphic function with simple poles of [[residue (complex analysis)|residue]] 1 at all positive integers. With notation as above, letting <math>p_k = 1/(z-k)</math> and <math>E = \mathbb{Z}^+</math>, Mittag-Leffler's theorem asserts (non-constructively) the existence of a meromorphic function <math>f</math> with principal part <math> p_k(z) </math> at <math>z=k</math> for each positive integer <math> k</math>. This <math>f </math> has the desired properties. More constructively we can let <math>f(z) = z\sum_{k=1}^\infty \frac{1}{k(z-k)} </math>. This series [[Normal convergence|converges normally]] on <math> \mathbb{C} </math> (as can be shown using the [[Weierstrass M-test|M-test]]) to a meromorphic function with the desired properties.
| |
| | |
| Another example is provided by
| |
| :<math>\frac \pi{\sin \pi z}=\frac{1}{z} + \sum_{k\in\mathbb{Z}, \, k \ne 0}(-1)^k\left(\frac{1}
| |
| {z-k}+\frac{1}{k}\right).</math>
| |
| | |
| == See also==
| |
| * [[Riemann-Roch theorem]]
| |
| *[[Weierstrass factorization theorem]]
| |
| * [[Liouville's theorem (complex analysis)|Liouville's theorem]]
| |
| | |
| ==References==
| |
| *{{citation|first=Lars|last=Ahlfors|authorlink=Lars Ahlfors|title=Complex analysis|publisher=McGraw Hill|year=1953|publication-date=1979|edition=3rd|isbn=0-07-000657-1}}.
| |
| *{{citation|first=John B.|last=Conway|title=Functions of One Complex Variable I|publisher=Springer-Verlag|year=1978|publication-date=1978|edition=2nd|isbn=0-387-90328-3}}.
| |
| | |
| ==External links==
| |
| * {{springer|title=Mittag-Leffler theorem|id=p/m064170}}
| |
| * {{planetmath reference|title=Mittag-Leffler's theorem|id=3732}}
| |
| | |
| [[Category:Theorems in complex analysis]]
| |
The individual who wrote the post is called Jayson Hirano and he completely digs that title. phone psychic readings (just click the next post) For years she's been operating as a journey agent. My wife and I live in Mississippi but now I'm contemplating other options. As a woman what she really likes is fashion and she's been performing it for quite a while.
My web site free tarot readings - free psychic readings (myfusionprofits.com)