Functional square root

In mathematics, a half iterate (sometimes called a functional square root) 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. For example, f(x) = 2x² is a functional square root of g(x) = 8x⁴. Similarly, the functional square root of the Chebyshev polynomials g(x) = T_n(x) is f(x) = cos (√n arccos(x)) , in general not a polynomial.

One notation that expresses that f is a functional square root of g is f = g_½.

The functional square root of the exponential function was studied by H. Kneser in 1950.

The solutions of f(f(x)) = x over the real numbers (the involutions of the reals) were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation.

A systematic procedure to produce arbitrary functional n-roots (including, beyond n= ½, continuous, negative, and infinitesimal n) relies on Schröder's equation.

Example

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From the general pedagogy web-site.   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 four integral iterates below it, starting with the second iterate (red). The green envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function.

See also

• Schröder's equation
• Iterated function
• Flow (mathematics)