# Euler's Formula, or a Study in Cool

## 25 November 2013

Let $i=\sqrt{-1}$, and let $e^{x}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}$.1 If follows that

Utilizing series expansion, this can be equivalently expressed this as

Because the series is absolutely convergent, the terms can be rearranged so to be expressed as

By recognizing the Maclaurin series for $\cos x$ and $\sin x$,2 we can collapse the series to

This is called Euler’s Formula,3 and perhaps one of the coolest things I’ve seen in mathematics. It establishes a relationship between exponentiation ($e^x$) and trigonometric functions - even if it is just imaginary.