Let , and let .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 and ,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 () and trigonometric functions - even if it is just imaginary.