NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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Imaginary Exponents
We may define imaginary exponents the same way that all sufficiently smooth real-valued functions of a real variable are generalized to the complex case--using Taylor series. A Taylor series expansion is just a polynomial (possibly of infinitely high order), and polynomials involve only addition, multiplication, and division. Since these elementary operations are also defined for complex numbers, any smooth function of a real variable
may be generalized to a function of a complex variable
by simply substituting the complex variable
for the real variable
in the Taylor series expansion.
Let
, where
is any positive real number. The Taylor series expansion expansion about
(''Maclaurin series''), generalized to the complex case is then
which is well defined (although we should make sure the series converges for every finite). We have
, so the first term is no problem. But what is
? In other words, what is the derivative of
at
? Once we find the successive derivatives of
for any complex
