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In complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the real natural logarithm ln ''x'' is the inverse of the real exponential function ''e''''x''. Thus, a logarithm of a complex number ''z'' is a complex number ''w'' such that ''e''''w'' = ''z''.〔Sarason, Section IV.9.〕 The notation for such a ''w'' is ln ''z'' or log ''z''. Since every nonzero complex number ''z'' has infinitely many logarithms,〔 care is required to give such notation an unambiguous meaning. If ''z'' = ''re''''iθ'' with ''r'' > 0 (polar form), then ''w'' = ln ''r'' + ''iθ'' is one logarithm of ''z;'' adding integer multiples of 2''πi'' gives all the others.〔 ==Problems with inverting the complex exponential function== For a function to have an inverse, it must map distinct values to distinct values, i.e., be injective. But the complex exponential function is not injective, because for any ''w'', since adding ''iθ'' to ''w'' has the effect of rotating ''e''''w'' counterclockwise ''θ'' radians. Even worse, the infinitely many numbers : forming a sequence of equally spaced points along a vertical line, are all mapped to the same number by the exponential function. So the exponential function does not have an inverse function in the standard sense.〔Conway, p. 39.〕〔Another interpretation of this is that the "inverse" of the complex exponential function is a multivalued function taking each nonzero complex number ''z'' to the ''set'' of all logarithms of ''z''.〕 There are two solutions to this problem. One is to restrict the domain of the exponential function to a region that ''does not contain any two numbers differing by an integer multiple of 2πi'': this leads naturally to the definition of branches of , which are certain functions that single out one logarithm of each number in their domains. This is analogous to the definition of on as the inverse of the restriction of to the interval : there are infinitely many real numbers ''θ'' with , but one (somewhat arbitrarily) chooses the one in . Another way to resolve the indeterminacy is to view the logarithm as a function whose domain is not a region in the complex plane, but a Riemann surface that ''covers'' the punctured complex plane in an infinite-to-1 way. Branches have the advantage that they can be evaluated at complex numbers. On the other hand, the function on the Riemann surface is elegant in that it packages together ''all'' branches of and does not require an arbitrary choice as part of its definition. 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「Complex logarithm」の詳細全文を読む スポンサード リンク
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