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Let's say that $f(x)=f^{1}(x)$ and that $f(f(x))=f^{2}(x)$. Moreover, $f^{n}(x)$ is the n-th iterate of $f(x)$, for $n \in \mathbb{N}$. I'm curious about extending iteration to larger number sets. For $n \in \mathbb{R}$, there's the concept flow (I think?). I don't understand the Wikipedia-article on this subject very well, though. I was hoping for some nice, concrete examples of iterated functions extended to the real or even complex numbers with which I might understand things better. If we take $f(x) = x^2 +3$, for example, what would $f^{\sqrt(2)}(x)$ be? Or, even more ambitiously, say that $g(x)=e^x$ How do we find $g^{\pi^2 + 3i}(x)$?

Thanks,

Max Muller

Editorial to the moderators: perhaps this should be CW?

  • 0
    Converted to CW. Since this is a big-list question.2010-12-17
  • 0
    @ Willie Wong: thanks.2010-12-17

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