**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## Back to e^(j theta)

We've now defined for any positive real number and any complex number . Setting and gives us the special case we need for Euler's identity. Since is its own derivative, the Taylor series expansion for for is the simplest series there could be:

The simplicity comes about because for all and because we chose to expand about the point . We of course define

Note that all even order terms are real while all odd order terms are imaginary. Separating out the real and imaginary parts gives

Comparing the Maclaurin expansion for with that of and proves Euler's identity. Recall that

so that

Plugging into the general Maclaurin series gives

Separating the Maclaurin expansion for into its even and odd terms (real and imaginary parts) gives

thus proving Euler's identity.