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^{ix}is periodic, with period of 2 pi, we can use the Fourier series...

[itex]\displaystyle \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{ix} \ dx + \frac{1}{\pi}\sum_{n=1}^{\infty}\left[\left(\int_{-\pi}^{\pi} e^{ix} cos(nx) \ dx \right) cos(nx) + \left(\int_{-\pi}^{\pi} e^{ix} sin(nx) \ dx\right) sin(nx)\right] = \frac{0}{2\pi} + \frac{1}{\pi}(\pi cos{x} + i\pi sin{x}) = cos{x} + i sin{x}[/itex]

...to describe e

^{ix}, right?

Isn't this an easier way to prove Euler's formula than using Taylor expansion? Or...am I missing something?