Functional Square Root

A functional square root of a function g is another function f that, when composed with itself, reproduces g: applying f twice to an input gives the same result as applying g once. For example, the function f of x equal to 2x squared is a functional square root of g of x equal to 8x to the fourth power. The idea traces back to Charles Babbage's 1815 study of involutions, functions that return their input unchanged after being applied twice, and was extended in 1950 by Hellmuth Kneser, who found a functional square root of the exponential function and laid groundwork for extending repeated function application to non-integer numbers of steps. Not every function has a functional square root of a simple form: a functional square root of a Chebyshev polynomial exists but need not be a polynomial, and functional square roots are typically found by solving a related equation known as Schröder's equation, which can admit infinitely many solutions once the domain is extended widely enough. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/