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Complex analysis:

* EQN:e^{ix}=cos(x)+i.sin(x)

In the special case EQN:x=\pi this reduces to Euler's Identity.

* EQN:e^{i\theta}=\cos(\theta)+i.\sin(\theta)

where /e/ is Euler's Number: 2.71828...

In the special case EQN:\theta=\pi this reduces to Euler's Identity.

The functions /cos/ and /sin/ come originally from trigonometry, and

it's just a little bit magical that they turn up in a purely algebraic

context.

Interestingly, the formula also means that in some cases you can take

logarithms of negative numbers. Specifically:

| EQN:\log(z)=\log(r.e^{i\theta})=\log(r)+i.\theta |

The log is not unique, though, because the same number /z/ is

represented by infinitely many values of EQN:\theta.