De Moivre's Theorem Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


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.


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]