In my research, I often use functional iteration, and I usually denote $$(f\circ f\circ f\circ\dots\circ f)(x)$$ by writing $$f^n(x)$$ where $f$ was composed with itself $n$ times. However, this notation can be confusing at times, especially when dealing with the trigonometric functions. Does anyone know of any elegant ways of expressing iterated function composition that do not conflict with preexisting notations?
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Parcly Taxel
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Franklin Pezzuti Dyer
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To future authors reading this question: Please use $(\sin x)^2$ and $(\log x)^2$ instead of $\sin^2 x$ and $\log^2 x$. The latter notation is ambiguous. The former is not. – user76284 Nov 03 '19 at 18:51
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@user76284 I dunno, I think it’s not a bad idea to write $\sin^2 x$ because authors almost always mean $(\sin x)^2$ and almost never $\sin\sin x$. Also, $\sin^2 x$ avoids those ugly parentheses that go around $(\sin x)^2$. – Franklin Pezzuti Dyer Nov 03 '19 at 20:13
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The problem is that $f^n$ normally denotes $f$ composed $n$ times. The "special notation" for trigonometric and logarithmic functions conflicts with that. You can see the conflict at play with $\sin^{-1} x$. Does it denote $\arcsin x$, or $(\sin x)^{-1}$? Why is the power of 2 a special case? – user76284 Nov 03 '19 at 20:24
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$\sin \sin x$ isn't common, but in number theory you do frequently see terms like $\log \log \log x$, which can be written more concisely as $\log^3 x$. – user76284 Nov 03 '19 at 20:30
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From Wikipedia:
To avoid ambiguity, some mathematicians choose to write $f^{\circ n}$ for the $n$-th iterate of the function $f$.
Parcly Taxel
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1And this convention makes sense in general. E.g. $A^n = ∏{i < n} A$ but $A^{\oplus n} = \bigoplus{i < n} A$. – user87690 Sep 24 '17 at 12:48
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Hint: The paper Notations for Iteration of Functions, Iteral by Valerii Salov might be interesting. He proposes a new notation and gives in section 3 a short survey about existing notation.
He refers in this section besides some others to
- Iterative Functional Equations by M. Kuczma, B. Choczewski, R. Ger stating that composition $\circ$ of functions is the only inner operation which be defined in the family $\mathcal{F}(x)$ of self-mappings for a set $X$. The system $(\mathcal{F}(X),\circ)$ is with $\mathrm{id}_X$ a non-commutative monoid with iterates defined as the powers of $n$ of an element $f\in\mathcal{F}(X)$, where $n$ is a non-negative integer: \begin{align*} n\in\mathbb{N},f^0=\mathrm{id}_X,f^{n+1}=f\circ f^n \end{align*}
another notation was introduced by D. Knuth in
- The Art of Computer Programming, Vol. 2 Seminumerical algorithms. There he chooses square parentheses to reduce ambiguity with powers of multiplication. \begin{align*} f^{[n]}(x)=f(f^{[n-1]}(x)) \end{align*}
Markus Scheuer
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