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Arctangent Series Expansion

For example, the arctangent function used above can be expanded as

$\displaystyle \arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots

Note that all even-order terms are zero. This is always the case for odd functions, i.e., functions satisfying $ f(-x)=-f(x)$ . For any smooth function, the odd-order terms of its Taylor expansion comprise the odd part of the function, while the even-order terms comprise the even part. The original function is clearly given by the sum of its odd and even parts.7.17

The clipping nonlinearity in Eq.$ \,$ (6.17) is not so amenable to a series expansion. In fact, it is its own series expansion! Since it is not differentiable at $ x=\pm1$ , it must be represented as three separate series over the intervals $ (-\infty,1]$ , $ [-1,1]$ , and $ [1,\infty)$ , and the result obtained over these intervals is precisely the definition of $ f(x)$ in Eq.$ \,$ (6.17).

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2017-02-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University