Series Expansions

Any ``smooth'' function can be expanded as a Taylor series expansion:

where ``smooth'' means that derivatives of all orders must exist over the range of validity. Derivatives of all orders are obviously needed at by the above expansion, and for the expansion to be valid everywhere, the function must be smooth for all .

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.