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

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

$\blacksquare$

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.