TITLE: What is your favorite proof that $e^{ix}$ has a period of $2\pi$?
QUESTION [10 upvotes]: 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.

REPLY [11 votes]: My favorite has always been Walter Rudin's proof in the prologue to his "Real and Complex Analysis" (2nd Ed.).  Here's a sketch:

Define $\exp$ in terms of the power series.
By manipulating the series, deduce that $\exp$ is a homomorphism from the additive group to the group of complex units.
Show it satisfies the usual first order ODE.
Define $\cos z$ and $\sin z$ as the real and imaginary parts of $\exp(iz)$, respectively.
Define $\pi$ as twice the smallest positive real root of $\cos$.
Deduce that $\exp( i \pi / 2) = i$.
By multiplying, conclude that $2 \pi i$ is a period of $\exp$.
Show, by means of the preceding properties, that no smaller period exists.