Asymptotic analysis of the derivatives of the inverse error function

Abstract

The inverse of the error function, $\operatorname{inverf}(x),$ has applications in diffusion problems, chemical potentials, ultrasound imaging, etc. We analyze the derivatives $\frac{d^{n}}{dz^{n}} \operatorname*{inverf}(z) |_{z=0}$, as $n\to \infty$ using nested derivatives and a discrete ray method. We obtain a very good approximation of $\operatorname{inverf}(x)$ through a high-order Taylor expansion around $x=0$. We give numerical results showing the accuracy of our formulas.