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Series Expansions

Any ``smooth'' function $ f(x)$ can be expanded as a Taylor series expansion:

$\displaystyle f(x) = f(0) + \frac{f^\prime(0)}{1}(x) + \frac{f^{\prime\prime}(0)}{1\cdot 2}(x)^2 + \frac{f^{\prime\prime\prime}(0)}{1\cdot 2\cdot 3}(x)^3 + \cdots,$ (7.18)

where ``smooth'' means that derivatives of all orders must exist over the range of validity. Derivatives of all orders are obviously needed at $ x=0$ by the above expansion, and for the expansion to be valid everywhere, the function $ f(x)$ must be smooth for all $ x$ .


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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