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, 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, 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. 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. 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)} .
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], although this is not unique, the opposite - rin being a solution of sin = rin ∘ rin, too. sin[1/4](x) = qin(x) [black curve above the orange curve] sin[–1](x) = arcsin(x) [dashed curve] Using this extension, sin[1/2](1) can be shown to be approximately equal to 0.90871. (See. For the notation, see [1] Archived 2022-12-05 at the Wayback Machine.)
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