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If we iterate the function $f(x) = \ln(x + 1)$, we get: $$f(f(x)) = f^2(x) = \ln(\ln(x + 1) + 1)$$ $$f(f(f(x))) = f^3(x) = \ln(\ln(\ln(x + 1) + 1) + 1)$$ $$f(f(f(f(x)))) = f^4(x) = \ln(\ln(\ln(\ln(x + 1) + 1) + 1) + 1)$$ And so forth. I was wondering if there is some clever way to deal with these forms of expressions to simplify them any more. Everything I tried failed (and spectacularly at that), so if there is some neat trick we could use to write the general iterate $f^n(x)$ in closed form.

Honestly, I don't expect any simplification to be possible, but I just wanted to be absolutely sure.

Daccache
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  • I dont think there is any simplification either – M.S.E Sep 03 '14 at 17:19
  • I do not know whether ther is, but the notation $f^n$ you used seems perfectly fine, since it is a widely used one. And I never came across a simplification (or it was a really special function). – flawr Sep 03 '14 at 17:20
  • I agree that there is no simple closed form. I think there might be a reasonable expression for the $n^{th}$ iterate in terms of a power series, which is nice if you happen to be studying the iteration of $f$ in a neighborhood of the origin. – Mark McClure Sep 03 '14 at 17:43
  • With regards to power series of the iterated log, this answer of mine may be of interest. @MarkMcClure – Semiclassical Sep 03 '14 at 17:52

1 Answers1

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The super-logarithm and the iterated logarithm function could be interesting, but these are kind of "limits" for your $f^{(n)}$. I doubt if there is simplification for a given $n$.

user153012
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