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Mathematics of the DFT
    Proof of Euler's Identity
       e^(j theta)


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e^(j theta)

We've now defined $ a^z$ for any positive real number $ a$ and any complex number $ z$. Setting $ a=e$ and $ z=j\theta$ gives us the special case we need for Euler's identity. Since $ e^x$ is its own derivative, the Taylor series expansion for $ f(x)=e^x$ is one of the simplest imaginable infinite series:

$\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots $

The simplicity comes about because $ f^{(n)}(0)=1$ for all $ n$ and because we chose to expand about the point $ x=0$. We of course define

$\displaystyle e^{j\theta} \isdef \sum_{n=0}^\infty \frac{(j\theta)^n}{n!} = 1 + j\theta - \frac{\theta^2}{2} - j\frac{\theta^3}{3!} + \cdots \,. $

Note that all even order terms are real while all odd order terms are imaginary. Separating out the real and imaginary parts gives

\begin{eqnarray*} \mbox{re}\left\{e^{j\theta}\right\} &=& 1 - \theta^2/2 + \thet... ...heta}\right\} &=& \theta - \theta^3/3! + \theta^5/5! - \cdots\,. \end{eqnarray*}

Comparing the Maclaurin expansion for $ e^{j\theta }$ with that of $ \cos(\theta)$ and $ \sin(\theta)$ proves Euler's identity. Recall from introductory calculus that

\begin{eqnarray*} \frac{d}{d\theta}\cos(\theta) &=& -\sin(\theta) \\ [5pt] \frac{d}{d\theta}\sin(\theta) &=& \cos(\theta) \end{eqnarray*}

so that

\begin{eqnarray*} \left.\frac{d^n}{d\theta^n}\cos(\theta)\right\vert _{\theta=0}... ...} \\ [5pt] 0, & n\;\mbox{\small even}. \\ \end{array} \right. \end{eqnarray*}

Plugging into the general Maclaurin series gives

\begin{eqnarray*} \cos(\theta) &=& \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}\theta... ...mbox{\tiny$n$\ odd}}}^\infty \frac{(-1)^{(n-1)/2}}{n!} \theta^n. \end{eqnarray*}

Separating the Maclaurin expansion for $ e^{j\theta }$ into its even and odd terms (real and imaginary parts) gives

\begin{eqnarray*} e^{j\theta} \isdef \sum_{n=0}^\infty \frac{(j\theta)^n}{n!} &... ...-1)^{(n-1)/2}}{n!} \theta^n\\ &=& \cos(\theta) + j\sin(\theta) \end{eqnarray*}

thus proving Euler's identity.

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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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