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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 Euler function ： ウィキペディア英語版 Euler function :''For other meanings, see List of topics named after Leonhard Euler''. In mathematics, the Euler function is given by :$\backslash phi(q)=\backslash prod\_^\backslash infty\; (1-q^k).$ Named after Leonhard Euler, it is a prototypical example of a q-series, a modular form, and provides the prototypical example of a relation between combinatorics and complex analysis. ==Properties== The coefficient $p(k)$ in the formal power series expansion for $1/\backslash phi(q)$ gives the number of all partitions of k. That is, :$\backslash frac=\backslash sum\_^\backslash infty\; p(k)\; q^k$ where $p(k)$ is the partition function of k. The Euler identity, also known as the Pentagonal number theorem is :$\backslash phi(q)=\backslash sum\_^\backslash infty\; (-1)^n\; q^.$ Note that $(3n^2-n)/2$ is a pentagonal number. The Euler function is related to the Dedekind eta function through a Ramanujan identity as :$\backslash phi(q)=\; q^\}\; \backslash eta(\backslash tau)$ where $q=e^$ is the square of the nome. Note that both functions have the symmetry of the modular group. The Euler function may be expressed as a Q-Pochhammer symbol: :$\backslash phi(q)=(q;q)\_\backslash infty$ The logarithm of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded about q=0, yielding: :$\backslash ln(\backslash phi(q))=-\backslash sum\_^\backslash infty\backslash frac\backslash ,\backslash frac$ which is a Lambert series with coefficients ''-1/n''. The logarithm of the Euler function may therefore be expressed as: :$\backslash ln(\backslash phi(q))=\backslash sum\_^\backslash infty\; b\_n\; q^n$ where :$b\_n=-\backslash sum\_\backslash frac=$ -(3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ... ) (see OEIS (A000203 )) On account of the following identity, :$\backslash sum\_\; d\; =\; \backslash sum\_\; \backslash frac\; n\; d$ this may also be written as :$\backslash ln(\backslash phi(q))=-\backslash sum\_^\backslash infty\; \backslash frac\; \backslash sum\_\; d$ 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「Euler function」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.