Generalised Euler-Jacobi Inversion Formula and Asymptotics beyond All Orders: 214 (London Mathematical Society Lecture Note Series, Series Number 214)

By considering special exponential series arising in number theory, the authors derive the generalized Euler-Jacobi series, expressed in terms of hypergeometric series. They then employ Dingle's theory of terminants to show how the divergences in both dominant and subdominant series of a complete asymptotic expansion can be tamed. The authors use numerical results to show that a complete asymptotic expansion can be made to agree with exact results for the generalized Euler-Jacobi series to any desired degree of accuracy.

In many real physical problems an exact solution cannot be obtained, and only approximations called asymptotic expansions exist. This work presents exciting new developments, previously neglected, in understanding the subdominant exponential terms of these expansions.