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https://en.wikipedia.org/wiki/Exponentiation#Iterated_functions https://en.wikipedia.org/wiki/Function_composition#Functional_powers https://calculus.subwiki.org/wiki/Higher_derivative

Why does exponentiation and iterated functions have similar notation as well as names? Are these two operations similar in any way? If yes, then what are some key points of differences and similarities between the two?

I'm aware of the fact that other notations for iterated functions exist too, but that doesn't seem to reason why the common notation for exponentiation is also the same for iterated functions as well as higher order derivatives.

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  • $\begingroup$ Well, if you write function application without brackets, notationally it is exactly the same. $f f f f x = f^4 x$, whether juxtaposition means function application or multiplication. $\endgroup$
    – Zhen Lin
    Commented Dec 23, 2021 at 3:50

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Mathematicians like compact notation for many reasons (ease of writing and pattern recognition being the most important ones).

Unfortunately, there are not that many compact ways of combining a number and a function. So if you are in an area where you need iterated functions frequently, why would you restrict yourself from using $f^n$ just because other parts of math use that notation for exponentiation (and vice versa)?

With a bit of experience, usually, no confusion will arise. For one, if you want to iterate a function, its domain and codomain need to agree, and if you want to exponentiate, the codomain needs to have some kind of multiplication. Often, only one of these will be the case and there is only one interpretation. In all other situations, context (and good surrounding text!) is key.

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  • $\begingroup$ Agreed, notation is the problem. Describing the context in such cases should help disambiguate b/w the two. I am curious to know, how does one distinguish between iteration and exponentiation in case of a function which satisfies both the conditions? Do they become one and the same in those cases? $\endgroup$ Commented Dec 24, 2021 at 5:16
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    $\begingroup$ The two are almost never the same, so you need to know which one it is. Again, in many cases only one interpretation even works. Then there are some examples where both would work but only one is common (e.g. $\sin^2 x$ is always $(\sin x)^2$); learning these comes with experience. If the right interpretation is not immediately obvious to the intended audience, the writing is bad. This happens. If you want to figure out what is meant in these cases, you have to look at the derivation or the use of the formula to find a place where only one interpretation works. $\endgroup$ Commented Dec 24, 2021 at 11:08

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