Functional square root

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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 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 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.[2] 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.[3][4] [5]

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.


Iterates of the sine function (blue), in the first half-period.     Half-iterate (orange), i.e., the sine's functional square root; the functional square root of that, the quarter-iterate (black) above it; and four integral iterates below it, starting with the second iterate (red). The green envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function. From the general pedagogy web-site.[6]
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) [not shown. would be above the green curve.]

See also[edit]


  1. ^ Kneser, H. (1950). "Reelle analytische Lösungen der Gleichung φ(φ(x)) = ex und verwandter Funktionalgleichungen". Journal fur die reine und angewandte Mathematik 187: 56–67. 
  2. ^ Jeremy Gray and Karen Parshall (2007) Episodes in the History of Modern Algebra (1800–1950), American Mathematical Society, ISBN 978-0-8218-4343-7
  3. ^ Schröder, E. (1870). "Ueber iterirte Functionen". Mathematische Annalen 3 (2): 296–322. doi:10.1007/BF01443992. 
  4. ^ Szekeres, G. (1958). "Regular iteration of real and complex functions". Acta Mathematica 100 (3–4): 361–376. doi:10.1007/BF02559539. 
  5. ^ Curtright, T.; Zachos, C. (2011). "Approximate solutions of functional equations". Journal of Physics A 44 (40): 405205. doi:10.1088/1751-8113/44/40/405205. 
  6. ^ Curtright, T.L. Evolution surfaces and Schröder functional methods.