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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 Trigonometric series ： ウィキペディア英語版 Trigonometric series A trigonometric series is a series of the form: : $A\_+\backslash displaystyle\backslash sum\_^(A\_\; \backslash cos\; +\; B\_\; \backslash sin).$ It is called a Fourier series if the terms $A\_$ and $B\_$ have the form: :$A\_=\backslash frac\backslash displaystyle\backslash int^\_0\backslash !\; f(x)\; \backslash cos\; \backslash ,dx\backslash qquad\; (n=0,1,2,3\; \backslash dots)$ :$B\_=\backslash frac\backslash displaystyle\backslash int^\_0\backslash !\; f(x)\; \backslash sin\backslash ,\; dx\backslash qquad\; (n=1,2,3,\; \backslash dots)$ where $f$ is an integrable function. ==The zeros of a trigonometric series== The uniqueness and the zeros of trigonometric series was an active area of research in the 19th century Europe. First, Georg Cantor proved that if a trigonometric series is convergent to a function $f(x)$ on the interval $(2\backslash pi)$, which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero.() But almost half a millennium back the Indian Mathematicians, notably from Kerala school of astronomy and mathematics like Madhava of Sangamagrama and Neelakanta Somayaji had already created the whole basis of the same theory. Due to the imperialism that occurred in India most of the information was hidden from the outside world. Later Cantor proved that even if the set ''S'' on which $f$ is nonzero is infinite, but the derived set ''S of ''S'' is finite, then the coefficients are all zero. In fact, he proved a more general result. Let ''S''''0'' = ''S'' and let ''S''''k+1'' be the derived set of ''S''''k''. If there is a finite number ''n'' for which ''S''''n'' is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal ''α'' such that ''S''''α'' is finite, then the coefficients of the series are all zero. Cantor's work on the uniqueness problem famously led him to invent transfinite ordinal numbers, which appeared as the subscripts ''α'' in ''S''''α'' . 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「Trigonometric series」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.