# Euler's Formula

## Theorem

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

Then:

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

where:

- $e^{i z}$ denotes the complex exponential function
- $\cos z$ denotes the complex cosine function
- $\sin z$ denotes complex sine function
- $i$ denotes the imaginary unit.

### Real Domain

This result is often presented and proved separately for arguments in the real domain:

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

Then:

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

### Corollary

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

## Proof

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

\(\ds \cos z + i \sin z\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {z^{2 n} } {\paren {2 n}!} + i \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {z^{2 n + 1} } {\paren {2 n + 1}!}\) | Definition of Complex Cosine Function and Definition of Complex Sine Function | |||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {\paren {-1}^n \dfrac {z^{2 n} } {\paren {2 n}!} + i \paren {-1}^n \dfrac {z^{2 n + 1} } {\paren {2 n + 1}!} }\) | Sum of Absolutely Convergent Series | |||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {\dfrac {\paren {i z}^{2 n} } {\paren {2 n}!} + \dfrac {\paren {i z}^{2 n + 1} } {\paren {2 n + 1}!} }\) | Definition of Imaginary Unit | |||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \dfrac {\paren {i z}^n} {n!}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds e^{i z}\) | Definition of Complex Exponential Function |

$\blacksquare$

## Examples

### Example: $e^{i \pi / 4}$

- $e^{i \pi / 4} = \dfrac {1 + i} {\sqrt 2}$

### Example: $e^{i \pi / 2}$

- $e^{i \pi / 2} = i$

### Example: $e^{-i \pi / 2}$

- $e^{-i \pi / 2} = -i$

### Example: $e^{i \pi}$

- $e^{i \pi} = -1$

### Example: $e^{2 i \pi}$

- $e^{2 i \pi} = 1$

### Example: $e^{2 k i \pi}$

- $\forall k \in \Z: e^{2 k i \pi} = 1$

## Also known as

**Euler's formula** in this and its corollary form are also found referred to as **Euler's identities**, but this term is also used for the specific example:

- $e^{i \pi} + 1 = 0$

It is wise when referring to it by name, therefore, to ensure that the equation itself is also specified.

## Also see

## Source of Name

This entry was named for Leonhard Paul Euler.

## Historical Note

Leonhard Paul Euler famously published what is now known as Euler's Formula in $1748$.

However, it needs to be noted that Roger Cotes first introduced it in $1714$, in the form:

- $\map \ln {\cos \theta + i \sin \theta} = i \theta$

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Euler's formula**