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

Result R425 on D786: Standard natural complex exponential function
Subresult of R5125: Euler's formulas

Euler's formulas for a real variable

Formulation 0

Let $y \in \mathbb{R}$ be a D993: Real number.
Let $i$ be the D371: Imaginary number.

Then

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

Formulation 1

Let $y \in \mathbb{R}$ be a D993: Real number.
Let $i$ be the D371: Imaginary number.

Then

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

Subresults

▶	R424: Euler's identity

Proofs

Proof 1

Let $y \in \mathbb{R}$ be a D993: Real number.
Let $i$ be the D371: Imaginary number.

This result is a particular case of R5125: Euler's formulas. $\square$