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
[edit]Notations expressing that f is a functional square root of g are f = g[1/2] and f = g1/2.[citation needed]
History
[edit]- 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
[edit]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
[edit]- 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 .
![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Sine_iterations.svg/680px-Sine_iterations.svg.png)
- 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]
- 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
[edit]References
[edit]- ^ 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.
- ^ 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. S2CID 116998358.
- ^ 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. S2CID 119142727.
- ^ Curtright, T. L. Evolution surfaces and Schröder functional methods Archived 2014-10-30 at the Wayback Machine.