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Functional square root

Function that, applied twice, gives another function

Wikipedia contributorsAdapted for Fathomly · View authors
2 min read
Source updated September 10, 2026

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.

01Notation

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.

Iterates of the sine function (blue), in the first half-period. Half-iterate (orange), i.e., the sine's functional square root; the functional square root of that, the quarter-iterate (black) above it, and further fractional iterates up to the 1/64th iterate. The functions below sine are six integral iterates below it, starting with the second iterate (red) and ending with the 64th iterate. The green envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function. The dashed line is the negative first iterate, i.e. the inverse of sine (arcsin).
Iterates of the sine function (blue), in the first half-period. Half-iterate (orange), i.e., the sine's functional square root; the functional square root of that, the quarter-iterate (black) above it, and further fractional iterates up to the 1/64th iterate. The functions below sine are six integral iterates below it, starting with the second iterate (red) and ending with the 64th iterate. The green envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function. The dashed line is the negative first iterate, i.e. the inverse of sine (arcsin).

02History

03Solutions

A systematic procedure to produce arbitrary functional n-roots (including arbitrary real, negative, and infinitesimal n) of functions 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.

04Examples

  • 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), 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], 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 Archived 2022-12-05 at the Wayback Machine.)

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Sources and credits

This article is adapted from the Wikipedia article “Functional square root”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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