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. 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 relies on the solutions of Schröder's equation. 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) = Tn(x), is f(x) = cos( arccos(x)), which in general is not a polynomial. f(x) = x/( + x(1 − )) is a functional square root of g(x) = x/(2 − x). sin2 = sin(sin(x)) [red curve] sin1 = sin(x) = rin(rin(x)) [blue curve] sin1/2 = rin(x) = qin(qin(x)) [orange curve] sin1/4 = qin(x) [black curve above the orange curve] sin–1 = arcsin(x) [dashed curve] (See. For the notation, see .