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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Mittag-Leffler's theorem ： ウィキペディア英語版 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 $\backslash mathbb\; C$ and $E\; \backslash subset\; D$ 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 $a\; \backslash in\; E$, 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 $f(z)\; =\; \backslash sum\_\; p\_a(z)$. If $E$ is not finite, consider the finite sum $S\_F(z)\; =\; \backslash sum\_\; p\_a(z)$ 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 ''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 :$p\_k\; =\; \backslash frac$ and $E\; =\; \backslash mathbb^+$, 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 :$f(z)\; =\; z\backslash sum\_^\backslash infty\; \backslash frac$. This series converges normally on $\backslash mathbb$ (as can be shown using the M-test) to a meromorphic function with the desired properties. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mittag-Leffler's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.