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In mathematics, a half iterate (sometimes called a functional square root) is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function is a function satisfying for all . *For example, is a functional square root of . *Similarly, the functional square root of the Chebyshev polynomials is , in general ''not a polynomial''. *Likewise, is a functional square root of . Notations expressing that is a functional square root of are and . *The functional square root of the exponential function was studied by Hellmuth Kneser in 1950.〔 〕 *The solutions of over ℝ (the involutions of the real numbers) were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation.〔Jeremy Gray and Karen Parshall (2007) ''Episodes in the History of Modern Algebra (1800–1950)'', American Mathematical Society, ISBN 978-0-8218-4343-7〕 A particular solution is for ; it includes = 0, or else . Babbage noted that for any given solution , its functional conjugate by an arbitrary invertible function is also a solution. A systematic procedure to produce ''arbitrary'' functional -roots (including, beyond , continuous, negative, and infinitesimal ) relies on the solutions of Schröder's equation. ==Example== : (style="color:red">red curve ) : (style="color:blue">blue curve ) : (style="color:orange">orange curve ) : (curve above the orange curve ) : (curve ) (Cf. the general pedagogy web-site.〔Curtright, T.L. (Evolution surfaces and Schröder functional methods. )〕) 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「functional square root」の詳細全文を読む スポンサード リンク
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