# Iterated self-composition of arbitrary function

Does there exist some notation that represents the iterative composition of a single-input, single-output function with itself? As in, say, $f_5(x)=f(f(f(f(f(x)))))$.

In other words, going by the above (incorrect, I'm pretty sure) notation:

$f(x)=x+1$, $f_n(m)=m+n$

I'm looking for the correct way to express the notion of "$f_n(x)$" for any $f$.

## 2 Answers

I've used $f^n$ when there is no confusion with $f(x)^n$.

I've seen $f^{\circ n}$ to emphasize composition.

• Perfect, thanks; the disambiguating notation is especially helpful. – zedoyle May 15 '14 at 2:06
• – lhf May 15 '14 at 10:29

For me, I use $f^n$ and also consistently never reuse this notation for $f(...)^n$, including things like $\cos(x)^2$. After all, it's more consistent to write $\cos(x)^2+\sin(x)^2=1$ than to give these functions additional special notation.