Generalised Euler-Jacobi Inversion Formula and Asymptotics beyond All Orders

Vic Kowalenko; N. E. Frankel; L. Glasser; T. Taucher

doi:10.1017/CBO9780511752513

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.

‘The book is of considerable value for the number theorist and for the analyst as well.’

Source: Monatshefte für Mathematik

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Generalised Euler-Jacobi Inversion Formula and Asymptotics beyond All Orders

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Contents

Frontmatter
pp i-iv

Preface
pp v-x

Acknowledgements
pp x-x

Contents
pp xi-xii

1 - Introduction
pp 1-3

2 - Exact Evaluation of Srp/q(a)
pp 3-12

3 - Properties of Sp/q(a)
pp 12-20

4 - Steepest Descent
pp 20-24

5 - Special Cases of Sp/q(a) for p/q < 2
pp 24-43

6 - Integer cases for Sp/q(a) where 2 < p/q < 7
pp 44-73

7 - Asymptotics beyond all Orders
pp 73-95

8 - Numerics for Terminant Sums
pp 96-105

9 - Conclusion
pp 105-114

References
pp 115-116

Tables
pp 117-126

Index
pp 127-129

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Generalised Euler-Jacobi Inversion Formula and Asymptotics beyond All Orders

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