Half-iterate

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 = g_1/2.

A systematic procedure to produce arbitrary functional n-roots (including, beyond n = 1/2, continuous, negative, and infinitesimal n) relies on the solutions of Schröder's equation.

(Cf. the general pedagogy web-site. For the notation, see [1].)

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