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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 © 2019-07-30 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University