# 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

 $\ds \cos z + i \sin z$ $=$ $\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {z^{2 n} } {\paren {2 n}!} + i \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {z^{2 n + 1} } {\paren {2 n + 1}!}$ 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.

## 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:

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