# 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

As Sine Function is Absolutely Convergent and Cosine Function is Absolutely Convergent, we have:

\(\displaystyle \cos \theta + i \sin \theta\) | \(=\) | \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {\theta^{2 n} } {\paren {2 n}!} + i \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {\theta^{2 n + 1} } {\paren {2 n + 1}!}\) | 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 |

$\blacksquare$

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 2$. Geometrical Representations: $(2.18)$ - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Euler's Formula

- For a video presentation of the contents of this page, visit the Khan Academy.

- 2001: Christian Berg:
*Kompleks funktionsteori*: $\S 1.5$