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 $x$.

Notation
Notations expressing that $f$ is a functional square root of $g$ are $f = g_{[1/2]}$ and $f = g_{1/2}$.

History

 * 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 $$\mathbb{R}$$ (the involutions of the real numbers) were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation. 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 $$g: \mathbb{C}\rarr \mathbb{C}$$ 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.

Examples

 * $f(x) = 2x^{2}$ is a functional square root of $g(x) = 8x^{4}$.
 * A functional square root of the $n$th Chebyshev polynomial, $$g(x)=T_n(x)$$, is $$f(x) = \cos{(\sqrt{n}\arccos(x))}$$, which in general is not a polynomial.
 * $$f(x) = x / (\sqrt{2} + x(1-\sqrt{2}))$$ is a functional square root of $$g(x)=x / (2-x)$$.




 * $sin_{[2]}(x) = sin(sin(x))$ [ red curve]
 * $sin_{[1]}(x) = sin(x) = rin(rin(x))$ [ blue curve]
 * $sin_{[1⁄2]}(x) = rin(x) = qin(qin(x))$ [ orange curve]
 * $sin_{[1⁄4]}(x) = qin(x)$ [black curve above the orange curve]
 * $sin_{[–1]}(x) = arcsin(x)$ [dashed curve]

(See. For the notation, see .)