Functional square root: Difference between revisions
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*The solutions of {{math|1=''f''(''f''(''x'')) = ''x''}} over <math>\mathbb{R}</math> (the [[involution (mathematics)|involution]]s of the [[real number]]s) were first studied by [[Charles Babbage]] in 1815, and this equation is called Babbage's [[functional equation]].<ref>[[Jeremy Gray]] and [[Karen Parshall]] (2007) ''Episodes in the History of Modern Algebra (1800–1950)'', [[American Mathematical Society]], {{ISBN|978-0-8218-4343-7}}</ref> A particular solution is {{math|1=''f''(''x'') = (''b'' − ''x'')/(1 + ''cx'')}} for {{math|''bc'' ≠ −1}}. Babbage noted that for any given solution {{math|''f''}}, its [[Topological conjugacy|functional conjugate]] {{math|Ψ<sup>−1</sup>∘ ''f'' ∘{{space|hair}}Ψ}} by an arbitrary [[invertible function|invertible]] function {{math|Ψ}} is also a solution. In other words, the [[Group (mathematics)|group]] of all invertible functions on the real line [[Group action (mathematics)|acts]] on the |
*The solutions of {{math|1=''f''(''f''(''x'')) = ''x''}} over <math>\mathbb{R}</math> (the [[involution (mathematics)|involution]]s of the [[real number]]s) were first studied by [[Charles Babbage]] in 1815, and this equation is called Babbage's [[functional equation]].<ref>[[Jeremy Gray]] and [[Karen Parshall]] (2007) ''Episodes in the History of Modern Algebra (1800–1950)'', [[American Mathematical Society]], {{ISBN|978-0-8218-4343-7}}</ref> A particular solution is {{math|1=''f''(''x'') = (''b'' − ''x'')/(1 + ''cx'')}} for {{math|''bc'' ≠ −1}}. Babbage noted that for any given solution {{math|''f''}}, its [[Topological conjugacy|functional conjugate]] {{math|Ψ<sup>−1</sup>∘ ''f'' ∘{{space|hair}}Ψ}} by an arbitrary [[invertible function|invertible]] function {{math|Ψ}} is also a solution. In other words, the [[Group (mathematics)|group]] of all invertible functions on the real line [[Group action (mathematics)|acts]] on the subset consisting of solutions to Babbage's functional equation by [[conjugacy class|conjugation]]. |
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==Solutions== |
==Solutions== |
Revision as of 13:36, 3 November 2020
In mathematics, a functional square root (sometimes called a half iterate) 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.
Notation
Notations expressing that f is a functional square root of g are f = g[1/2] and f = g1/2.[citation needed]
History
- The functional square root of the exponential function (now known as a half-exponential function) was studied by Hellmuth Kneser in 1950.[1]
- The solutions of f(f(x)) = x over (the involutions of the real numbers) were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation.[2] A particular solution is f(x) = (b − x)/(1 + cx) for bc ≠ −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. In other words, the group of all invertible functions on the real line acts on the subset consisting of solutions to Babbage's functional equation by conjugation.
Solutions
A systematic procedure to produce arbitrary functional n-roots (including, beyond n = 1/2,[clarification needed] continuous, negative, and infinitesimal n) of functions g: ℂ→ℂ relies on the solutions of Schröder's equation.[3][4][5] Infinitely many trivial solutions exist when the domain a root function f is allowed to be sufficiently larger than that of g.
Examples
- f(x) = 2x2 is a functional square root of g(x) = 8x4.
- A functional square root of the nth Chebyshev polynomial, g(x) = Tn(x), is f(x) = cos(√n arccos(x)), which in general is not a polynomial.
- f(x) = x/(√2 + x(1 − √2)) is a functional square root of g(x) = x/(2 − x).
- sin[2](x) = sin(sin(x)) [red curve]
- sin[1](x) = sin(x) = rin(rin(x)) [blue curve]
- sin[½](x) = rin(x) = qin(qin(x)) [orange curve]
- sin[¼](x) = qin(x) [black curve above the orange curve]
- sin[–1](x) = arcsin(x) [dashed curve]
(See [6]. For the notation, see [1].)
See also
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References
- ^ Kneser, H. (1950). "Reelle analytische Lösungen der Gleichung φ(φ(x)) = ex und verwandter Funktionalgleichungen". Journal für die reine und angewandte Mathematik. 187: 56–67.
- ^ Jeremy Gray and Karen Parshall (2007) Episodes in the History of Modern Algebra (1800–1950), American Mathematical Society, ISBN 978-0-8218-4343-7
- ^ Schröder, E. (1870). "Ueber iterirte Functionen". Mathematische Annalen. 3 (2): 296–322. doi:10.1007/BF01443992.
- ^ Szekeres, G. (1958). "Regular iteration of real and complex functions". Acta Mathematica. 100 (3–4): 361–376. doi:10.1007/BF02559539.
- ^ Curtright, T.; Zachos, C.; Jin, X. (2011). "Approximate solutions of functional equations". Journal of Physics A. 44 (40): 405205. arXiv:1105.3664. Bibcode:2011JPhA...44N5205C. doi:10.1088/1751-8113/44/40/405205.
- ^ Curtright, T. L. Evolution surfaces and Schröder functional methods.