# Euler's Formula/Proof

## Theorem

Let $z \in \C$ be a complex number.

Then:

$e^{i z} = \cos z + i \sin z$

## Proof

 $\displaystyle \cos z + i \sin z$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {z^{2 n} } {\paren {2 n}!} + i \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {z^{2 n + 1} } {\paren {2 n + 1}!}$ Definition of Complex Cosine Function and Definition of Complex Sine Function $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty \paren {\paren {-1}^n \dfrac {z^{2 n} } {\paren {2 n}!} + i \paren {-1}^n \dfrac {z^{2 n + 1} } {\paren {2 n + 1}!} }$ Sum of Absolutely Convergent Series $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty \paren {\dfrac {\paren {i z}^{2 n} } {\paren {2 n}!} + \dfrac {\paren {i z}^{2 n + 1} } {\paren {2 n + 1}!} }$ Definition of Imaginary Unit $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty \dfrac {\paren {i z}^n} {n!}$ $\displaystyle$ $=$ $\displaystyle e^{i z}$ Definition of Complex Exponential Function

$\blacksquare$