Generalised Euler-Jacobi Inversion Formula and Asymptotics beyond All Orders

This work presents exciting new developments in understanding the subdominant exponential terms of asymptotic expansions which have previously been neglected.

Vic Kowalenko (Author), N. E. Frankel (Author), L. Glasser (Author), T. Taucher (Author)

9780521497985, Cambridge University Press

Paperback, published 14 September 1995

142 pages
22.9 x 15.2 x 0.8 cm, 0.22 kg

'The book is of considerable value for the number theorist and for the analyst as well.' Monatshefte für Mathematik

This work, first published in 1995, presents developments in understanding the subdominant exponential terms of asymptotic expansions which have previously been neglected. By considering special exponential series arising in number theory, the authors derive the generalised Euler-Jacobi series, expressed in terms of hypergeometric series. Dingle's theory of terminants is then employed to show how the divergences in both dominant and subdominant series of a complete asymptotic expansion can be tamed. Numerical results are used to illustrate that a complete asymptotic expansion can be made to agree with exact results for the generalised Euler-Jacobi series to any desired degree of accuracy. All researchers interested in the fascinating area of exponential asymptotics will find this a most valuable book.

1. Introduction
2. Exact evaluation of Srp/q(a)
3. Properties of Sp/q(a)
4. Steepest descent
5. Special cases of Sp/q(a) for p/q<2
6. Integer cases for Sp/q(a) where 2

7. Asymptotics beyond all orders
8. Numerics for terminant sums
9. Conclusion
References
Tables.

Subject Areas: Calculus & mathematical analysis [PBK]