NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 ''.'' (The right-hand side is assumed to be understood.) Since is just a particular number, we only really have to explain what we mean by imaginary exponents. (We'll also see where 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 , where is any real number, and is any positive real number.

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Subsections

• Positive Integer Exponents
• Properties of Exponents
• The Exponent Zero
• Negative Exponents
• Rational Exponents
  • Special Case: The Mth Roots of Unity
• Real Exponents
• A First Look at Taylor Series
• Imaginary Exponents
• Derivatives of f(x)=a^x
• Back to e
• Sidebar on Mathematica
• Back to e^(j theta)

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