# Euler's Formula/Proof

## Theorem

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

Then:

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

## Proof

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

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

$\blacksquare$

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(ii)}$: $(4.14)$

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

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