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In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that entire functions can be represented by a product involving their zeroes. In addition, every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence. The theorem is named after Karl Weierstrass. A second form of the theorem extends to meromorphic functions and allows one to consider a given meromorphic function as a product of three factors: terms depending on the function's poles and zeroes, and an associated non-zero holomorphic function. ==Motivation== The consequences of the fundamental theorem of algebra are twofold.〔.〕 Firstly, any finite sequence in the complex plane has an associated polynomial that has zeroes precisely at the points of that sequence, Secondly, any polynomial function in the complex plane has a factorization where ''a'' is a non-zero constant and ''c''''n'' are the zeroes of ''p''. The two forms of the Weierstrass factorization theorem can be thought of as extensions of the above to entire functions. The necessity of extra machinery is demonstrated when one considers the product if the sequence is not finite. It can never define an entire function, because the infinite product does not converge. Thus one cannot, in general, define an entire function from a sequence of prescribed zeroes or represent an entire function by its zeroes using the expressions yielded by the fundamental theorem of algebra. A necessary condition for convergence of the infinite product in question is that each factor must approach 1 as . So it stands to reason that one should seek a function that could be 0 at a prescribed point, yet remain near 1 when not at that point and furthermore introduce no more zeroes than those prescribed. Weierstrass' ''elementary factors'' have these properties and serve the same purpose as the factors above. 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「Weierstrass factorization theorem」の詳細全文を読む スポンサード リンク
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