Is fractional inverse of a function a known thing?

Question

I know there's fractional Fourier transform, fractional derivative, maybe some other transformations generalized from being discrete to continuous. Now I wonder if there's any way to generalize a function inversion to make it fractional.

What I mean is if $f(x)$ is an invertible function $g$, and $f^{-1}(x)$, as it is usually denoted, is inverse of $f$, then what would a function, satisfying the following, be:

$$g(g(x))=f^{-1}(x).$$

It seems natural to me to call it "half-inverse" of $f$ and denote like $f^{-\frac12}(x)$. Similarly would other fractional inverses (i.e. with another "order of inversion") be defined.

Is it a known thing? If yes, what would you recommend to read on its theory?

Answer 1

I think a more fundamental question would revolve around composition of functions. Once you define "fractional composition" your "fractional inverses" are covered. Look up this wiki: http://en.wikipedia.org/wiki/Function_composition

under functional powers to see how they define a functional square root.

This wiki has more on fractional iterations of functions and "flows": http://en.wikipedia.org/wiki/Iterated_function

Answer 2

The idea of fractional, negative, negative fractional, and even complex $a + b\sqrt{-1}$ iteration of functions was considered by several 19th century mathematicians. The following book is good place to begin if you're interested in the historical roots of this subject.

Daniel S. Alexander, A History of Complex Dynamics: From Schröder to Fatou and Julia, Aspects of Mathematics #E 24, Friedrich Vieweg and Sohn Verlagsgesellschaft mbH, 1994, viii + 165 pages.