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 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

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