ThmDex – An index of mathematical definitions, results, and conjectures.

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Euler's formulas

Formulation 0

Let

$z \in \mathbb{C}$ be a D1207: Complex number.
Let

Then

(1)	$\begin{equation} \exp(i z) = \cos z + i \sin z \end{equation}$
(2)	$\begin{equation} \exp(- i z) = \cos z - i \sin z \end{equation}$

Formulation 1

Let

$z \in \mathbb{C}$ be a D1207: Complex number.
Let

Then

(1)	$\begin{equation} e^{i z} = \cos z + i \sin z \end{equation}$
(2)	$\begin{equation} e^{- i z} = \cos z - i \sin z \end{equation}$

Also known as

Euler's formulae

Subresults

▶	R425: Euler's formulas for a real variable