**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## Positive Integer Exponents

The ''original'' definition of exponents which ''actually makes sense'' applies only to positive integer exponents:

where is real.Generalizing this definition involves first noting its abstract mathematical properties, and then making sure these properties are preserved in the generalization.