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[Posting some images here for possible future reuse.]

\displaystyle \lim_{n\to\infty}\left(1 + \frac{ix}{n}\right)^n = \cos x + i \sin x

A non-rigorous argument: when n is large enough so that x/n is small, (1 + ix/n) is roughly (hand-waving) the point on the unit circle at arc length (and hence angle) x/n:

So multiplication by (1+ix/n) roughly corresponds to rotation by angle x/n. Multiplication by (1+ix/n)^n, which is multiplication by (1+ix/n) n times, roughly corresponds to rotation by angle n(x/n) = x. As n \to \infty, “roughly” becomes exact.

Animation for x = 1:

Image generated from Python-generated SVG files; code available if anyone wants.

In particular, once one accepts the fact/definition that \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^n = e^x (for instance, show that the function f(x) = \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^n satisfies f(x+y) = f(x)f(y)), it is true that e^{i\pi} is a rotation by π, that is,

\displaystyle e^{i\pi} = -1

Written by S

Tue, 2011-06-21 at 18:04:25

Posted in mathematics

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