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In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all ''D'' ''except'' a set of isolated points (the poles of the function), at each of which the function must have a Laurent series. This terminology comes from the Ancient Greek ''meros'' (''μέρος''), meaning ''part'', as opposed to ''holos'' (''ὅλος''), meaning ''whole''. Every meromorphic function on ''D'' can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on ''D'': any pole must coincide with a zero of the denominator. Intuitively, a meromorphic function is a ratio of two well-behaved (holomorphic) functions. Such a function will still be well-behaved, except possibly at the points where the denominator of the fraction is zero. If the denominator has a zero at ''z'' and the numerator does not, then the value of the function will be infinite; if both parts have a zero at ''z'', then one must compare the multiplicities of these zeros. From an algebraic point of view, if ''D'' is connected, then the set of meromorphic functions is the field of fractions of the integral domain of the set of holomorphic functions. This is analogous to the relationship between the rational numbers and the integers. ==History== In the 1930s, in group theory, a ''meromorphic function'' (or ''meromorph'') was a function from a group ''G'' into itself that preserved the product on the group. The image of this function was called an ''automorphism'' of ''G''.〔Zassenhaus pp. 29, 41〕 Similarly, a ''homomorphic function'' (or ''homomorph'') was a function between groups that preserved the product, while a ''homomorphism'' was the image of a homomorph. This terminology is now obsolete. The term ''endomorphism'' is now used for the function itself, with no special name given to the image of the function. The term ''meromorph'' is no longer used in group theory. 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「Meromorphic function」の詳細全文を読む スポンサード リンク
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