Functional square root

Notations expressing that f is a functional square root of g are f = g[1/2] and f = g1/2[citation needed][dubious – discuss], or rather f = g 1/2 (see Iterated function#Fractional_iterates_and_flows,_and_negative_iterates), 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.

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.

[3][4][5] Infinitely many trivial solutions exist when the domain of a root function f is allowed to be sufficiently larger than that of g.

[6] For the notation, see [1] Archived 2022-12-05 at the Wayback Machine.)

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 ).