Euler's Formula

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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 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

$\blacksquare$


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.


Sources