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[dubious – discuss], 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.
History
- The functional square root of the exponential function (now known as a half-exponential function) was studied by Hellmuth Kneser in 1950,1 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.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 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.345 Infinitely many trivial solutions exist when the domain of 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 ) = 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)} .
Using this extension, sin[1/2](1) can be shown to be approximately equal to 0.90871.6
(See.7 For the notation, see [1] Archived 2022-12-05 at the Wayback Machine.)
See also
- Iterated function
- Function composition
- Abel equation
- Schröder's equation
- Flow (mathematics)
- Superfunction
- Fractional calculus
- Half-exponential function
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. doi:10.1515/crll.1950.187.56. S2CID 118114436. /wiki/Hellmuth_Kneser ↩
Jeremy Gray and Karen Parshall (2007) Episodes in the History of Modern Algebra (1800–1950), American Mathematical Society, ISBN 978-0-8218-4343-7 /wiki/Jeremy_Gray ↩
Schröder, E. (1870). "Ueber iterirte Functionen". Mathematische Annalen. 3 (2): 296–322. doi:10.1007/BF01443992. S2CID 116998358. /wiki/Ernst_Schr%C3%B6der_(mathematician) ↩
Szekeres, G. (1958). "Regular iteration of real and complex functions". Acta Mathematica. 100 (3–4): 361–376. doi:10.1007/BF02559539. /wiki/George_Szekeres ↩
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. /wiki/Thomas_Curtright ↩
https://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf https://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf ↩
Curtright, T. L. Evolution surfaces and Schröder functional methods Archived 2014-10-30 at the Wayback Machine. http://www.physics.miami.edu/~curtright/Schroeder.html ↩