Euler's Identity

Euler's identity (or ``theorem'' or ``formula'') is

To ``prove'' this, we will first define what we mean by `` ''. (The right-hand side, , is assumed to be understood.) Since is just a particular real 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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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.