Negative Exponents

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Negative Exponents

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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Negative Exponents

What should $a^{-1}$ be? Multiplying it by $a$ gives

\

Solving $a^{-1} \ for $a^{-1}$ then gives
\

Similarly, we obtain

\

for all integer values of $M$, i.e., $\.

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