Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Euler's Identity

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

$\displaystyle e^{j\theta} = \cos(\theta) + j\sin(\theta)
$   (Euler's Identity)

To ``prove'' this, we will first define what we mean by `` $ e^{j\theta }$ ''. (The right-hand side, $ \cos(\theta) +
j\sin(\theta)$ , is assumed to be understood.) Since $ e$ is just a particular real 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.


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``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 © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA