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<math> \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots </math> | <math> \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots </math> | ||
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| + | <math> e^{ix} = 1 + ix + /frac{{ix}^2}{2!} + /frac{{ix}^3}{3!} + /frac{{ix}^4}{4!} + /frac{{ix}^5}{5!} + \cdots </math> | ||
Revision as of 04:52, 5 September 2008
Euler's Forumla
$ e^{ix} = \cos x + i * \sin x $
Proof
Using Taylor Series:
$ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots $
$ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots $
$ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots $
$ e^{ix} = 1 + ix + /frac{{ix}^2}{2!} + /frac{{ix}^3}{3!} + /frac{{ix}^4}{4!} + /frac{{ix}^5}{5!} + \cdots $
