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 Identity
• Positive Integer Exponents
• Properties of Exponents
• The Exponent Zero
• Negative Exponents
• Rational Exponents
• Real Exponents
• A First Look at Taylor Series
• Imaginary Exponents
• Derivatives of f(x)=a^x
• Back to e
• e^(j theta)
• Back to Mth Roots
• Roots of Unity
• Direct Proof of De Moivre's Theorem
• Euler_Identity Problems

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.