## Series Expansions Reference

This is something that I use so often that I should have done it much sooner. Series expansions are a very useful tool that they don’t talk about nearly enough in college classes. Most of the time, for e.g. small signals, the first non-constant term is sufficient for an approximation – and instead of using inverse trigonometric functions or worse to solve your equations, you have a linear or maybe quadratic set of equations that is so much easier to handle. I can never remember the correct coefficients, though, and frequently look them up in Wikipedia. With this nice sheet tacked to the old office wall, these times will be gone.

One needs to be careful, however, not to let the function argument deviate too much for a series approximation to become invalid. That’s why I added graphs of the function itself (dashed black), the linear term (red), the quadratic term (blue) and the cubic term (green) in the vicinity of $x=0$. This way it should be much easier to estimate the validity of the various approximations.

• Marcus wrote:

Inverse Error Function

\begin{align} \mathrm{erf}^{-1} x &= \sum_{k=0}^\infty \frac{c_k}{2k+1} \left( \frac{\sqrt{\pi}}{2} x \right)^{2k+1} \\ &= \frac{\sqrt{\pi}}{2} \left( x + \frac{\pi}{12}x^3 + … \right) \end{align}

where

$$c_0 = 0 \quad \text{and} \quad c_k = \sum_{m=0}^{k-1} \frac{c_m c_{k-1-m}}{\left(m+1\right) \left(2m+1\right)}$$