If I have some composite function $g$, defined as $$g(x)=(k\:\circ\:k\:\circ\:...\:\circ\:k)$$ where $k(x)$ is essentially put into itself $n$ times, is there any compact notation I could express $g(x)$ with?
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2One common usage for iterated computations is to write $g(x) = k^n(x)$, acknowledging that this would normally be ambiguous but hopefully clear from the context. – WA Don Sep 10 '21 at 10:08
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The $k^n(x)$ mentioned in comments is probably the most common. Some people use $k^{(n)}(x)$ and I have also seen $k^{\circ n}(x)$. The last is not as common as the others, but is arguably the clearest of the three so as to distinguish from other possible meanings: $k^n$ could be understood as the $n$th power, and $k^{(n)}$ could mean the rising factorial or the $n$th derivative!
Wikipedia on Iterated function mentions two more: $k^{[n]}(x)$ and $\:{}^n\!k(x)$. I think they are rare, and would not recommend using them.
If $k^n(x)$ is clear enough in your context, you can use it. Otherwise, to be clear, use $k^{\circ n}(x)$.
Jukka Kohonen
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