NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

This chapter outlines the proof of Euler's Identity, which is an important tool for working with complex numbers. It is one of the critical elements of the DFT definition that we need to understand.

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Subsections

• Euler's Theorem
  • 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)
• Informal Derivation of Taylor Series Expansion
• Derivation of Taylor Series Expansion with Remainder
• Formal Statement of Taylor's Theorem
• The Weierstrass (Polynomial) Approximation Theorem
• Further Notes on Differentiability of Audio Signals

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