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Functional square root

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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

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Notations expressing that f is a functional square root of g are f = g[1/2] and f = g1/2[citation needed][dubiousdiscuss], or rather f = g 1/2 (see Iterated function#Fractional_iterates_and_flows,_and_negative_iterates), although this leaves the usual ambiguity with taking the function to that power in the multiplicative sense, just as f ² = f ∘ f can be misinterpreted as x ↦ f(x)².

History

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Solutions

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A systematic procedure to produce arbitrary functional n-roots (including arbitrary real, negative, and infinitesimal n) of functions relies on the solutions of Schröder's equation.[3][4][5] Infinitely many trivial solutions exist when the domain of a root function f is allowed to be sufficiently larger than that of g.

Examples

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  • f(x) = 2x2 is a functional square root of g(x) = 8x4.
  • A functional square root of the nth Chebyshev polynomial, , is , which in general is not a polynomial.
  • is a functional square root of .
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 further fractional iterates up to the 1/64th iterate. The functions below sine are six integral iterates below it, starting with the second iterate (red) and ending with the 64th iterate. The green envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function. The dashed line is the negative first iterate, i.e. the inverse of sine (arcsin).
sin[2](x) = sin(sin(x)) [red curve]
sin[1](x) = sin(x) = rin(rin(x)) [blue curve]
sin[1/2](x) = rin(x) = qin(qin(x)) [orange curve], although this is not unique, the opposite - rin being a solution of sin = rin ∘ rin, too.
sin[1/4](x) = qin(x) [black curve above the orange curve]
sin[–1](x) = arcsin(x) [dashed curve]

(See.[6] For the notation, see [1] Archived 2022-12-05 at the Wayback Machine.)

See also

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References

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  1. ^ 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. doi:10.1515/crll.1950.187.56. S2CID 118114436.
  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. S2CID 116998358.
  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.; 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. S2CID 119142727.
  6. ^ Curtright, T. L. Evolution surfaces and Schröder functional methods Archived 2014-10-30 at the Wayback Machine.