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.

Down­load the Series Expan­sions Cheat Sheet.

• 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

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