Complex analysis:

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

In the special case $\theta=\pi$ 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:


The log is not unique, though, because the same number z is represented by infinitely many values of $\theta.$

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