# Euler's Formula/Real Domain/Proof 5

## Theorem

Let $\theta \in \R$ be a real number.

Then:

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

## Proof

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

$\blacksquare$