De Moivre's Theorem

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. De Moivre's Theorem

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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De Moivre’s Theorem

As a more complicated example of the value of the polar form, we’ll proveDe Moivre’s theorem:

\

Working this out using sum-of-angle identities from trigonometry is laborious. However, using Euler’s identity, De Moivre’s theorem simply “falls out”:
\

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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