On approximations to a class of Jaeger integrals

Abstract

The object of this paper is to revisit a family of improper integrals first investigated by Jaeger (Jaeger 1942 Proc. R. Soc. Edinb., A 61 , 223–228). Of particular interest is a sub-class related to axisymmetric diffusion as it occurs in contemporary electrochemistry, specifically chronopotentiometry and chronoamperometry; applications that demand precision well beyond what can be achieved by simple interpolation of the tabulated solutions given by Jaeger and coworkers. To that end, the paper first outlines the less known numerical techniques necessary to solve such integrals and then employs them to obtain numerical solutions over a broad range of the temporal parameter τ , which includes the asymptotic approximations for small and large τ . Computed also are time integrals not previously calculated. More useful to practitioners, however, are approximations to the integrals that are easy to evaluate and sufficiently simple to be manipulated analytically, and the remainder of the work is devoted to such approximants. Each is constructed from a base function which captures the precise asymptotic behaviour at small and large τ plus a correction function, which is fitted either by a half-range Fourier sine series or an inverse power series in τ . Inclusion of sufficiently many terms in each series allows the approximants to realize an accuracy concordant with that of the numerical solutions. The approximants may also be integrated with respect to τ to obtain appropriate time integrals for use in convolution algorithms.