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In mathematics, the Koenigs function is a function arising in complex analysis and dynamical systems. Introduced in 1884 by the French mathematician Gabriel Koenigs, it gives a canonical representation as dilations of a univalent holomorphic mapping, or a semigroup of mappings, of the unit disk in the complex numbers into itself. ==Existence and uniqueness of Koenigs function== Let ''D'' be the unit disk in the complex numbers. Let be a holomorphic function mapping ''D'' into itself, fixing the point 0, with not identically 0 and not an automorphism of ''D'', i.e. a Möbius transformation defined by a matrix in SU(1,1). By the Denjoy-Wolff theorem, leaves invariant each disk |''z'' | < ''r'' and the iterates of converge uniformly on compacta to 0: in fact for 0 < < 1, : for |''z'' | ≤ ''r'' with ''M''(''r'' ) < 1. Moreover '(0) = with 0 < || < 1. proved that there is a unique holomorphic function ''h'' defined on ''D'', called the Koenigs function, such that (0) = 0, '(0) = 1 and Schröder's equation is satisfied, : The function ''h'' is ''the uniform limit on compacta of the normalized iterates'', . Moreover, if is univalent, so is . As a consequence, when (and hence ) are univalent, can be identified with the open domain . Under this conformal identification, the mapping becomes multiplication by , a dilation on . 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「Koenigs function」の詳細全文を読む スポンサード リンク
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