as a function of a real variable, apparently. Part of the freedom in choosing a proof is that you get to choose what definition of $e^{ix}$ to start from -- do you use a differential equation? a power series? a definition in terms of trig functions? Another bit of freedom is that you get to choose what definition of $\pi$ to start from.
What is your favorite proof that $e^{ix}$ has a period of $2\pi$?
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calculus
soft-question
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1Even if x is complex, $\exp(ix)$ is still periodic. – 2010-09-02
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0No takers for using the ODE?! – 2010-09-02
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1Maybe I didn't really frame the question clearly enough, but these are all kind of boring proofs. Perhaps I'd get some more interesting responses on MathOvervlow? – 2010-09-03
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0I suppose the recognition that e^ix = cosx + isinx from Taylor series is really surprising when you first see it. But I find it inelegant because you're reasoning about an infinite number of terms (granted, they're really simple terms) in order to understand something about a finite number of functions. I kind of assumed there'd be another proof out there that doesn't rely so heavily on Taylor series... – 2010-09-03
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0great example of how stupid the moderators are. This could have been an interesting discussion – 2018-12-28