I think you are looking for the functional square root operation: https://en.wikipedia.org/wiki/Functional_square_root https://en.wikipedia.org/wiki/Functional_

I think you are looking for the functional square root operation: https://en.wikipedia.org/wiki/Functional_square_root https://en.wikipedia.org/wiki/Functional_square_root 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. For your example of f(f(x)) = e^x, that would be the https://en.wikipedia.org/wiki/Half-exponential_function https://en.wikipedia.org/wiki/Half-exponential_function with a=1 and b=e. Unfortunately there's no simple arithmetic representation of f(x).