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This is a newbie question, but I would be grateful for any reference that you could give.

let $f(x) \in \mathbb{A}$, where $x \in \mathbb{A}$

Is there a symbol to indicate the repeated application of $f$ unto its results, in a manner similar to $\sum$ and $\prod$?

Thank you!

Sklivvz
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  • I believe this is a dupe. Unfortunately I am not able to find the previous one. – Aryabhata Nov 15 '10 at 20:06
  • Additionally: how is the study of repeated application of functions called? E.g. the limit as n goes to infinity of $f^n(x)$ – Sklivvz Nov 15 '10 at 20:07
  • @Moron, if it's a dupe do close it, I wasn't able to find it either though :-/ – Sklivvz Nov 15 '10 at 20:08
  • @Sklivvz: The question of what happens with $f^n(x)$ when $n$ goes to infinity in very complicated in general. See, for instance, http://en.wikipedia.org/wiki/Mandelbrot_set. On the other hand, sometimes the situation is simpler: http://en.wikipedia.org/wiki/Banach_fixed_point_theorem – Rasmus Nov 15 '10 at 20:32
  • @Rasmus, indeed. Actually, I did study that theorem 15 years ago... I had completely forgotten about that (IANAM) :-) – Sklivvz Nov 15 '10 at 20:41
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    @Moron: I believe you are thinking of this question: http://math.stackexchange.com/questions/8111/a-short-way-to-say-ffffx – Hans Lundmark Nov 15 '10 at 21:22
  • @Hans: Yes that is it. – Aryabhata Nov 15 '10 at 21:24
  • Technically, though, I was asking if there is a symbol like $\sum$ or $\prod$... which is a different question as the one mentioned. – Sklivvz Nov 15 '10 at 21:26
  • @Sklivv: Yes, It didn't seem an exact dupe and that is why I didn't cast my close vote :-) – Aryabhata Nov 15 '10 at 21:30

2 Answers2

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It is usually indicated by $f\circ f$ or if there are more than two repeated applications, simply $f^n(x)$

Timothy Wagner
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5

The symbol $\circ$ is used for composition of functions: $g\circ h(x)$ means $g(h(x))$. For composing a function with itself, many people use $f^{\circ n}(x)$, defined inductively by \begin{align*} f^{\circ 1}(x) &= f(x),\\ f^{\circ(k+1)}(x) &= f\Bigl( f^{\circ k}(x)\Bigr). \end{align*} So $f^{\circ 2}(x) = f(f(x))$; $f^{\circ 3}(x) = f(f^{\circ 2}(x)) = f(f(f(x)))$, etc.

Timothy Wagner also notes that $f^n(x)$ is often used; that is true, but it has the disadvantage of often being ambiguous. For example, $\sin^2(x)$ is (almost) always understood to mean $(\sin(x))^2$, and not $\sin(\sin(x))$.

Arturo Magidin
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