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Direct Proof of De Moivre's Theorem

In §2.10, De Moivre's theorem was introduced as a consequence of Euler's identity:

$\displaystyle \zbox {\left[\cos(\theta) + j \sin(\theta)\right] ^n =
\cos(n\theta) + j \sin(n\theta), \qquad\hbox{for all $n\in\mathbb{R}$}}

To provide some further insight into the ``mechanics'' of Euler's identity, we'll provide here a direct proof of De Moivre's theorem for integer $ n$ using mathematical induction and elementary trigonometric identities.

Proof: To establish the ``basis'' of our mathematical induction proof, we may simply observe that De Moivre's theorem is trivially true for $ n=1$ . Now assume that De Moivre's theorem is true for some positive integer $ n$ . Then we must show that this implies it is also true for $ n+1$ , i.e.,

$\displaystyle \left[\cos(\theta) + j \sin(\theta)\right] ^{n+1} = \cos[(n+1)\theta] + j \sin[(n+1)\theta]. \protect$ (3.2)

Since it is true by hypothesis that

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

multiplying both sides by $ [\cos(\theta) + j \sin(\theta)]$ yields
$\displaystyle \left[\cos(\theta) + j \sin(\theta)\right] ^{n+1}$ $\displaystyle =$ $\displaystyle \left[\cos(n\theta) + j \sin(n\theta)\right]
\left[\cos(\theta) + j \sin(\theta)\right]$  
  $\displaystyle =$ $\displaystyle \qquad\!
\left[\cos(n\theta)\cos(\theta) -\sin(n\theta)\sin(\theta)\right]$  
    $\displaystyle \,+\, j \left[\sin(n\theta)\cos(\theta)+\cos(n\theta)\sin(\theta)\right].
\protect$ (3.3)

From trigonometry, we have the following sum-of-angle identities:

\sin(\alpha+\beta) &=& \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\\
\cos(\alpha+\beta) &=& \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)

These identities can be proved using only arguments from classical geometry.3.8Applying these to the right-hand side of Eq.(3.3), with $ \alpha=n\theta$ and $ \beta=\theta$ , gives Eq.(3.2), and so the induction step is proved. $ \Box$

De Moivre's theorem establishes that integer powers of $ [\cos(\theta) + j \sin(\theta)]$ lie on a circle of radius 1 (since $ \cos^2(\phi)+\sin^2(\phi)=1$ , for all $ \phi\in[-\pi,\pi]$ ). It therefore can be used to determine all $ N$ of the $ N$ th roots of unity (see §3.12 above). However, no definition of $ e$ emerges readily from De Moivre's theorem, nor does it establish a definition for imaginary exponents (which we defined using Taylor series expansion in §3.7 above).

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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 © 2018-02-13 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University