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In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in the 1830s and 1840s, places an important restriction on antiderivatives that can be expressed as elementary functions. The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. A standard example of such a function is whose antiderivative is (with a multiplier of a constant) the error function, familiar from statistics. Other examples include the functions and . Liouville's theorem states that elementary antiderivatives, if they exist, must be in the same differential field as the function, plus possibly a finite number of logarithms. == Definitions == For any differential field ''F'', there is a subfield : Con(''F'') = , called the constants of ''F''. Given two differential fields ''F'' and ''G'', ''G'' is called a logarithmic extension of ''F'' if ''G'' is a simple transcendental extension of ''F'' (i.e. ''G'' = ''F''(''t'') for some transcendental ''t'') such that : ''Dt'' = ''Ds''/''s'' for some ''s'' in ''F''. This has the form of a logarithmic derivative. Intuitively, one may think of ''t'' as the logarithm of some element ''s'' of ''F'', in which case, this condition is analogous to the ordinary chain rule. But it must be remembered that ''F'' is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to ''F''. Similarly, an exponential extension is a simple transcendental extension that satisfies : ''Dt'' = ''t'' ''Ds''. With the above caveat in mind, this element may be thought of as an exponential of an element ''s'' of ''F''. Finally, ''G'' is called an elementary differential extension of ''F'' if there is a finite chain of subfields from ''F'' to ''G'' where each extension in the chain is either algebraic, logarithmic, or exponential. 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「Liouville's theorem (differential algebra)」の詳細全文を読む スポンサード リンク
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