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In mathematics, an elementary function is a function of one variable which is the composition of a finite number of arithmetic operations , exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of ''n''th roots). The elementary functions include the trigonometric and hyperbolic functions and their inverses, as they are expressible with complex exponentials and logarithms. It follows directly from the definition that the set of elementary functions is closed under arithmetic operations and composition. It is also closed under differentiation. It is not closed under limits and infinite sums. Elementary functions are analytic at all but a finite number of points. Importantly, the elementary functions are ''not'' closed under integration. The Liouvillian functions are defined as the elementary functions and, recursively, the integrals of the Liouvillian functions. Some elementary functions, such as roots, logarithms, or inverse trigonometric functions, are not entire functions and may be multivalued. Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841. An algebraic treatment of elementary functions was started by Joseph Fels Ritt in the 1930s. ==Examples== Examples of elementary functions include: * Addition, e.g. (''x''+1) * Multiplication, e.g. (2''x'') * and * The last function is equal to the inverse cosine trigonometric function in the entire complex domain. Hence, is an elementary function. An example of a function that is ''not'' elementary is the error function * a fact that may not be immediately obvious, but can be proven using the Risch algorithm. 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「Elementary function」の詳細全文を読む スポンサード リンク
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