Asymptotic analysis of the derivatives of the inverse error function

TL;DR

This paper provides an asymptotic analysis of the derivatives of the inverse error function at zero, enabling high-accuracy Taylor expansions useful in various scientific applications.

Contribution

It introduces a novel asymptotic approach using nested derivatives and a discrete ray method to analyze high-order derivatives of the inverse error function.

Findings

Derived accurate asymptotic formulas for derivatives at zero

Developed high-order Taylor expansions of the inverse error function

Validated formulas with numerical results showing high accuracy

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.