De Moivre's Theorem

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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,

can be any real number, not just an integer.

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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