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In mathematics, the Lambert W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function ''f''(''z'') = ''ze''''z'' where ''e''''z'' is the exponential function and ''z'' is any complex number. In other words : By substituting in the above equation, we get the defining equation for the W function (and for the W relation in general): : for any complex number ''z. Since the function ''ƒ'' is not injective, the relation ''W'' is multivalued (except at 0). If we restrict attention to real-valued ''W'', the complex variable ''z'' is then replaced by the real variable ''x'', and the relation is defined only for ''x'' ≥ −1/''e'', and is double-valued on (−1/''e'', 0). The additional constraint ''W'' ≥ −1 defines a single-valued function ''W''0(''x''). We have ''W''0(0) = 0 and ''W''0(−1/''e'') = −1. Meanwhile, the lower branch has ''W'' ≤ −1 and is denoted ''W''−1(''x''). It decreases from ''W''−1(−1/''e'') = −1 to ''W''−1(0−) = −∞. The Lambert ''W'' relation cannot be expressed in terms of elementary functions.〔.〕 It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. the maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions) and also occurs in the solution of delay differential equations, such as ''y(''t'') = ''a'' ''y''(''t'' − 1). In biochemistry, and in particular enzyme kinetics, a closed-form solution for the time course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function. : ==Terminology== The Lambert ''W''-function is named after Johann Heinrich Lambert. The main branch ''W''0 is denoted by ''Wp'' in the Digital Library of Mathematical Functions and the branch ''W''−1 is denoted by ''Wm'' there. The notation convention chosen here (with ''W''0 and ''W''−1) follows the canonical reference on the Lambert-''W'' function by Corless, Gonnet, Hare, Jeffrey and Knuth. 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「Lambert W function」の詳細全文を読む スポンサード リンク
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