Arctangent Example

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

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

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.^S.1

The clipping nonlinearity in Eq.$\,$(S.1) 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 obtain over these intervals is precisely the definition of $f(x)$ in Eq.$\,$(S.1).