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Euler-Moivre-Formula

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The exponential function with imaginary argument can be expressed in terms of the trigonometric functions:

$\displaystyle \exp (\mathrm{i} t) = \cos t + \mathrm{i} \sin t $

for $ t \in \mathbb{R}$. Hence, the cosine and the sine correspond to the real and the imaginary part of complex numbers with unit modulus ( $ \vert\exp(\mathrm{i}t)\vert=1$).

Inverting the the above formula

$\displaystyle \cos t$ $\displaystyle = \mathrm{Re}\; e^{\mathrm{i} t} = \frac{1}{2}\left( e^{\mathrm{i} t}+ e^{-\mathrm{i} t} \right)$    
$\displaystyle \sin t$ $\displaystyle = \mathrm{Im} \;e^{\mathrm{i} t} = \frac{1}{2 \mathrm{i}}\left( e^{\mathrm{i} t}-e^{-\mathrm{i} t} \right) \,.$    

The identities, relating $ \exp$, $ \cos$, and $ \sin$, are due to Euler and Moivre. They form the basis for the geometric interpretation of complex numbers and play an important role in Fourier analysis.

(Authors: Höllig/Kopf/Abele)

Example:

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  automatically generated 4/ 7/2008