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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

• History

• Solutions

• Examples

• See also

• References

Notations expressing that f is a functional square root of g are f = g[1/2] and f = g1/2, or rather f = g 1/2 (see Iterated Function), 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.

• The functional square root of the exponential function (now known as a half-exponential function) was studied by Hellmuth Kneser in 1950,cite-ref-sqrtexp-1-0[1] later providing the basis for extending tetration to non-integer heights in 2017.

• The solutions of f(f(x)) = x over R {\displaystyle \mathbb {R} } (the involutions of the real numbers) were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation.cite-ref-2[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.

A systematic procedure to produce arbitrary functional n-roots (including arbitrary real, negative, and infinitesimal n) of functions g : C → → C {\displaystyle g:\mathbb {C} \rightarrow \mathbb {C} } relies on the solutions of Schröder's equation.cite-ref-schr-3-0[3]cite-ref-4[4]cite-ref-5[5] Infinitely many trivial solutions exist when the domain of a root function f is allowed to be sufficiently larger than that of g.

• f(x) = 2x2 is a functional square root of g(x) = 8x4.

• A functional square root of the nth Chebyshev polynomial, g ( x ) = T n ( x ) {\displaystyle g(x)=T_{n}(x)} , is f ( x ) = cos ⁡ ⁡ ( n arccos ⁡ ⁡ ( x ) ) {\displaystyle f(x)=\cos {({\sqrt {n}}\arccos(x))}} , which in general is not a polynomial.

• f ( x ) = x / ( 2 + x ( 1 − − 2 ) ) {\displaystyle f(x)=x/({\sqrt {2}}+x(1-{\sqrt {2}}))} is a functional square root of g ( x ) = x / ( 2 − − x ) {\displaystyle g(x)=x/(2-x)} .

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]

Using this extension, sin[⁠1/2⁠](1) can be shown to be approximately equal to 0.90871.cite-ref-6[6]

(See.cite-ref-7[7] For the notation, see [1] Archived 2022-12-05 at the Wayback Machine.)

• Iterated function

• Function composition

• Abel equation

• Schröder's equation

• Flow (mathematics)

• Superfunction

• Fractional calculus

• Half-exponential function

cite-note-sqrtexp-11. ↑ citerefkneser-h-1950Kneser, 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.

cite-note-22. ↑ Jeremy Gray and Karen Parshall (2007) Episodes in the History of Modern Algebra (1800–1950), American Mathematical Society, ISBN 978-0-8218-4343-7

cite-note-schr-33. ↑ citerefschr-der-e-1870Schröder, E. (1870). "Ueber iterirte Functionen". Mathematische Annalen. 3 (2): 296–322. doi:10.1007/BF01443992. S2CID 116998358.

cite-note-44. ↑ citerefszekeres-g-1958Szekeres, G. (1958). "Regular iteration of real and complex functions". Acta Mathematica. 100 (3–4): 361–376. doi:10.1007/BF02559539.

cite-note-55. ↑ citerefcurtright-t-zachos-c-jin-x-2011Curtright, 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.

cite-note-66. ↑ https://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf

cite-note-77. ↑ Curtright, T. L. Evolution surfaces and Schröder functional methods Archived 2014-10-30 at the Wayback Machine.