I wonder what a function f(x) with the property that f(f(x)) = e^x might look like. (My usual caveat: DON’T TELL ME & RUIN THE WONDERING STAGE.)
I'm not sure, but whatever it is has to solve this nonlinear differential equationpic.twitter.com/E0CxIFwjLR
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Maybe my question is naive{?} but: Does that boxed equation assume that f'(x) is well-defined i.e. that f is a differentiable function?
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Indeed that is an unjustified assumption I am making, which I hope to--but am too lazy atm to-- "verify" by numerical plots
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I’ll solve this numerically tomorrow and see if it looks like anything I recognize, and/or if curve fitting to the solution gives something correct. I don’t see any reason f can’t be holomorphic though, and I’m pretty sure Picard-Lindelöf gives us existence and uniqueness
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In particular, the derivative is never zero and everywhere continuous, so we also have a guarantee that f is invertible (by whatever that theorem is called that says a function is locally invertible if its derivative is continuous and nonzero in that neighborhood)
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In mathematics, it is almost too easy (& interesting) to put together some symbols & numbers in a valid way, while the task of interpreting the meaning of these bunch of symbols can be daunting.
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