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The Exponential Function

\begin{eqnarray*}
\mrr {e^x}{\isdef }{\displaystyle\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n}%
{e^x}{=}{\displaystyle\sum_{n=0}^\infty\frac{x^n}{n!}}
\mr {e^{j\theta}}{\cos(\theta)+j\sin(\theta)}%
{e^{jn\theta}}{\cos(n\theta)+j\sin(n\theta)}
\\ [10pt]
\mr {\sin(\theta)}{\frac{e^{j\theta}-e^{-j\theta}}{2j}}%
{\cos(\theta)}{\frac{e^{j\theta}+e^{-j\theta}}{2}}
\mrr {e}{=}{2.7\,1828\,1828\,4590\,\ldots}{}{}{}
\end{eqnarray*}


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA