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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 Jacobi theta functions (notational variations) ： ウィキペディア英語版 Jacobi theta functions (notational variations) There are a number of notational systems for the Jacobi theta functions. The notations given in the Wikipedia article define the original function :$$ \vartheta_(z; \tau) = \sum_^\infty \exp (\pi i n^2 \tau + 2 \pi i n z) which is equivalent to :$$ \vartheta_(z, q) = \sum_^\infty q^ \exp (2 \pi i n z) However, a similar notation is defined somewhat differently in Whittaker and Watson, p. 487: :$$ \vartheta_(x) = \sum_^\infty q^ \exp (2 \pi i n x/a) This notation is attributed to "Hermite, H.J.S. Smith and some other mathematicians". They also define :$$ \vartheta_(x) = \sum_^\infty (-1)^n q^ \exp (\pi i (2 n + 1) x/a) This is a factor of ''i'' off from the definition of $\backslash vartheta\_$ as defined in the Wikipedia article. These definitions can be made at least proportional by ''x'' = ''za'', but other definitions cannot. Whittaker and Watson, Abramowitz and Stegun, and Gradshteyn and Ryzhik all follow Tannery and Molk, in which :$$ \vartheta_1(z) = -i \sum_^\infty (-1)^n q^ \exp ((2 n + 1) i z) :$$ \vartheta_2(z) = \sum_^\infty q^ \exp ((2 n + 1) i z) :$$ \vartheta_3(z) = \sum_^\infty q^ \exp (2 n i z) :$$ \vartheta_4(z) = \sum_^\infty (-1)^n q^ \exp (2 n i z) Note that there is no factor of π in the argument as in the previous definitions. Whittaker and Watson refer to still other definitions of $\backslash vartheta\_j$. The warning in Abramowitz and Stegun, "There is a bewildering variety of notations...in consulting books caution should be exercised," may be viewed as an understatement. In any expression, an occurrence of $\backslash vartheta(z)$ should not be assumed to have any particular definition. It is incumbent upon the author to state what definition of $\backslash vartheta(z)$ is intended. ==References== * Milton Abramowitz and Irene A. Stegun, ''Handbook of Mathematical Functions'', (1964) Dover Publications, New York. ISBN 0-486-61272-4. ''(See section 16.27ff.)'' * I. S. Gradshteyn and I. M. Ryzhik, ''Table of Integrals, Functions, and Products'', (1980) Academic Press, London. ISBN 0-12-294760-6. ''(See section 8.18)'' * E. T. Whittaker and G. N. Watson, ''A Course in Modern Analysis'', fourth edition, Cambridge University Press, 1927. ''(See chapter XXI for the history of Jacobi's θ functions)'' 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「Jacobi theta functions (notational variations)」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.