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In the mathematical field of complex analysis, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity of at ''z'' = 0. For a pole of the function ''f''(''z'') at point ''a'' the function approaches infinity as ''z'' approaches ''a''. == Definition == Formally, suppose ''U'' is an open subset of the complex plane C, ''p'' is an element of ''U'' and ''f'' : ''U'' \ → C is a function which is holomorphic over its domain. If there exists a holomorphic function ''g'' : ''U'' → C, such that ''g(p)'' is nonzero, and a positive integer ''n'', such that for all ''z'' in ''U'' \ : holds, then ''p'' is called a pole of ''f''. The smallest such ''n'' is called the order of the pole. A pole of order 1 is called a simple pole. A few authors allow the order of a pole to be zero, in which case a pole of order zero is either a regular point or a removable singularity. However, it is more usual to require the order of a pole to be positive. From above several equivalent characterizations can be deduced: If ''n'' is the order of pole ''p'', then necessarily ''g''(''p'') ≠ 0 for the function ''g'' in the above expression. So we can put : for some ''h'' that is holomorphic in an open neighborhood of ''p'' and has a zero of order ''n'' at ''p''. So informally one might say that poles occur as reciprocals of zeros of holomorphic functions. Also, by the holomorphy of ''g'', ''f'' can be expressed as: : This is a Laurent series with finite ''principal part''. The holomorphic function (on ''U'') is called the ''regular part'' of ''f''. So the point ''p'' is a pole of order ''n'' of ''f'' if and only if all the terms in the Laurent series expansion of ''f'' around ''p'' below degree −''n'' vanish and the term in degree −''n'' is not zero. 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「Pole (complex analysis)」の詳細全文を読む スポンサード リンク
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