Euler's Theorem

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Euler'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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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Euler’s Theorem

Euler’s Theorem (or “identity” or “formula”) is

\

To “prove” this, we must first define what we mean by “$e^{j\.” (The right-hand side is assumed to be understood.) Since $e$ is just a particular number, we only really have to explain what we mean by imaginary exponents. (We’ll also see where $e$ comes from in the process.) Imaginary exponents will be obtained as a generalization of real exponents. Therefore, our first task is to define exactly what we mean by $a^x$, where$x$ is any real number, and $a>0$ is any positive real number.



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