Functional square root
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.
Notations expressing that f is a functional square root of g are f = g[1/2] and f = g1/2.
- The functional square root of the exponential function (now known as a half-exponential function) was studied by Hellmuth Kneser in 1950.
- 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. 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 = 1/2, continuous, negative, and infinitesimal n) relies on the solutions of Schröder's equation.
- f(x) = 2x2 is a functional square root of g(x) = 8x4.
- the functional square root of the Chebyshev polynomials g(x) = Tn(x) is f(x) = cos(√ arccos(x)), in general not a polynomial.
- Likewise, f(x) = x/(√ + x(1 − √)) is a functional square root of g(x) = x/(2 − x).
- sin(x) = sin(sin(x)) [red curve]
- sin(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) [dashed curve]
- 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.
- 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.
- 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. doi:10.1088/1751-8113/44/40/405205.
- Curtright, T.L. Evolution surfaces and Schröder functional methods.