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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 + \theta^4/4! - \cdots \\ [5pt]
\mbox{im}\left\{e^{j\theta}\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}
&=& \left\{\begin{array}{ll}
(-1)^{n/2}, & n\;\mbox{\small even} \\ [5pt]
0, & n\;\mbox{\small odd} \\
\end{array} \right. \\ [10pt]
\left.\frac{d^n}{d\theta^n}\sin(\theta)\right\vert _{\theta=0}
&=& \left\{\begin{array}{ll}
(-1)^{(n-1)/2}, & n\;\mbox{\small odd} \\ [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^n \\
&=& \sum_{\stackrel{n\geq 0}{\vspace{2pt}\mbox{\tiny$n$\ even}}}^\infty \frac{(-1)^{n/2}}{n!} \theta^n \\
\sin(\theta) &=& \sum_{\stackrel{n\geq 0}{\vspace{2pt}\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!}
&=& \sum_{\stackrel{n\geq 0}{\vspace{2pt}\mbox{\tiny$n$\ even}}}^\infty \frac{(-1)^{n/2}}{n!} \theta^n
+ j \sum_{\stackrel{n\geq 0}{\vspace{2pt}\mbox{\tiny$n$\ odd}}}^\infty \frac{(-1)^{(n-1)/2}}{n!} \theta^n\\
&=& \cos(\theta) + j\sin(\theta)
\end{eqnarray*}

thus proving Euler's identity.


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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 © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA