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- Functional_square_root abstract "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 g is a function f satisfying f(f(x)) = g(x) for all x. For example, f(x) = 2x² is a functional square root of g(x) = 8x4. Similarly, the functional square root of the Chebyshev polynomials g(x) = Tn(x) is f(x) = cos (√n arccos(x)), in general not a polynomial.Notations expressing that f is a functional square root of g are f = g[½] and f = g½.The functional square root of the exponential function was studied by Hellmuth Kneser in 1950.The solutions of f(f(x)) = x over the real numbers (the involutions of the reals) were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation.A systematic procedure to produce arbitrary functional n-roots (including, beyond n= ½, continuous, negative, and infinitesimal n) relies on the solutions of Schröder's equation.Half-iterations (and other non-integer iterations) of derivation and integration are studied under fractional calculus. As with sin and arcsin, fractional and multiple derivatives and integrals and can be generalized into one function, differintegral.".
- Functional_square_root thumbnail SineIterates.jpg?width=300.
- Functional_square_root wikiPageID "7207519".
- Functional_square_root wikiPageRevisionID "602279776".
- Functional_square_root hasPhotoCollection Functional_square_root.
- Functional_square_root subject Category:Functional_analysis.
- Functional_square_root subject Category:Functional_equations.
- Functional_square_root comment "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 g is a function f satisfying f(f(x)) = g(x) for all x. For example, f(x) = 2x² is a functional square root of g(x) = 8x4.".
- Functional_square_root label "Functional square root".
- Functional_square_root label "Media iteración".
- Functional_square_root sameAs Media_iteración.
- Functional_square_root sameAs m.025w3wl.
- Functional_square_root sameAs Q3890203.
- Functional_square_root sameAs Q3890203.
- Functional_square_root wasDerivedFrom Functional_square_root?oldid=602279776.
- Functional_square_root depiction SineIterates.jpg.
- Functional_square_root isPrimaryTopicOf Functional_square_root.
