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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.Likewise, f(x) = x/(√2+x(1−√2)) is a functional square root of g(x) = x/(2−x). 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 ℝ (the involutions of the real numbers) were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation. A particular solution is f(x) = (b − x)/(1 + cx) for bc ≠ −1; it includes c = 0, or else |b| ≅ |c| ≫ 1. Babbage noted that for any given solution f, its functional conjugate Ψ−1 ○ f ○ Ψ by an arbitrary invertible function Ψ is also a solution.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.".
- Functional_square_root thumbnail Sine_iterations.svg?width=300.
- Functional_square_root wikiPageID "7207519".
- Functional_square_root wikiPageLength "5442".
- Functional_square_root wikiPageOutDegree "23".
- Functional_square_root wikiPageRevisionID "674639415".
- Functional_square_root wikiPageWikiLink Abel_equation.
- Functional_square_root wikiPageWikiLink Category:Functional_analysis.
- Functional_square_root wikiPageWikiLink Category:Functional_equations.
- Functional_square_root wikiPageWikiLink Charles_Babbage.
- Functional_square_root wikiPageWikiLink Chebyshev_polynomials.
- Functional_square_root wikiPageWikiLink Exponential_function.
- Functional_square_root wikiPageWikiLink Flow_(mathematics).
- Functional_square_root wikiPageWikiLink Fractional_calculus.
- Functional_square_root wikiPageWikiLink Function_(mathematics).
- Functional_square_root wikiPageWikiLink Function_composition.
- Functional_square_root wikiPageWikiLink Functional_equation.
- Functional_square_root wikiPageWikiLink Hellmuth_Kneser.
- Functional_square_root wikiPageWikiLink Involution_(mathematics).
- Functional_square_root wikiPageWikiLink Iterated_function.
- Functional_square_root wikiPageWikiLink Mathematics.
- Functional_square_root wikiPageWikiLink Schrxc3xb6ders_equation.
- Functional_square_root wikiPageWikiLink Square_root.
- Functional_square_root wikiPageWikiLink Superfunction.
- Functional_square_root wikiPageWikiLink Topological_conjugacy.
- Functional_square_root wikiPageWikiLink File:Sine_iterations.svg.
- Functional_square_root wikiPageWikiLinkText "Functional square root".
- Functional_square_root wikiPageWikiLinkText "functional square root".
- Functional_square_root wikiPageWikiLinkText "half iterate".
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- Functional_square_root subject Category:Functional_analysis.
- Functional_square_root subject Category:Functional_equations.
- Functional_square_root hypernym Root.
- Functional_square_root type Eukaryote.
- Functional_square_root type Function.
- 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 sameAs Q3890203.
- Functional_square_root sameAs Funkcia_kvadrata_radiko.
- Functional_square_root sameAs Media_iteración.
- Functional_square_root sameAs 函数的平方根.
- Functional_square_root sameAs m.025w3wl.
- Functional_square_root sameAs Q3890203.
- Functional_square_root wasDerivedFrom Functional_square_root?oldid=674639415.
- Functional_square_root depiction Sine_iterations.svg.
- Functional_square_root isPrimaryTopicOf Functional_square_root.