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A First Look at Taylor Series

Most ``smooth'' functions $ f(x)$ can be expanded in the form of a Taylor series expansion:

$\displaystyle f(x) = f(x_0) + \frac{f^\prime(x_0)}{1}(x-x_0)
+ \frac{f^{\prim...
+ \frac{f^{\prime\prime\prime}(x_0)}{1\cdot 2\cdot 3}(x-x_0)^3
+ \cdots .

This can be written more compactly as

$\displaystyle f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n,

where `$ n!$' is pronounced ``$ n$ factorial''. An informal derivation of this formula for $ x_0=0$ is given in Appendix E. Clearly, since many derivatives are involved, a Taylor series expansion is only possible when the function is so smooth that it can be differentiated again and again. Fortunately for us, all audio signals are in that category, because hearing is bandlimited to below $ 20$ kHz, and the audible spectrum of any sum of sinusoids is infinitely differentiable. (Recall that $ \sin^\prime(x)=\cos(x)$ and $ \cos^\prime(x)=-\sin(x)$, etc.). See §E.6 for more about this point.

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Next: Imaginary Exponents

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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 for details.


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