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De Moivre's Theorem

As a more complicated example of the value of the polar form, we'll prove De Moivre's theorem:

$\displaystyle \left[\cos(\theta) + j \sin(\theta)\right] ^n =
\cos(n\theta) + j \sin(n\theta)
$

Working this out using sum-of-angle identities from trigonometry is laborious (see §3.13 for details). However, using Euler's identity, De Moivre's theorem simply ``falls out'':

$\displaystyle \left[\cos(\theta) + j \sin(\theta)\right] ^n =
\left[e^{j\theta}\right] ^n = e^{j\theta n} =
\cos(n\theta) + j \sin(n\theta)
$

Moreover, by the power of the method used to show the result, $ n$ can be any real number, not just an integer.


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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