## Generalised Euler-Jacobi Inversion Formula and Asymptotics Beyond All OrdersThis 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. |

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accuracy algebraic series appearing in Eq asymp asymptotic forms asymptotic series becomes behaviour Berry and Howls Bessel functions Borel summation cancel coefficients column of Table complete asymptotic expansion contour divergent series equation exact terminant sums exponentially decaying finite functions in Eq gamma function generalised Euler-Jacobi series geometric functions geometry given by Eq Gradshteyn and Ryzhik Hence hyperasymptotic hypergeometric functions appearing integers integral in Eq Introducing Eqs L-asymptotics late terms Luke,s Meijer G-function Mellin-Barnes method of steepest number of terms obtained by Ramanujan optimal number oscillatory exponential p/q equal parameters path of steepest r.h.s. of Eq Ramanujan and Berndt Ramanujan-Berndt result recursion relation remainder result given Riemann zeta function Ryzhik 16 saddle point series for p/q steepest descent Stokes phenomenon subdominant exponential series subdominant terms tail technique terminant sums TkK(a totic utilising values of p/q yields zeta function zeta series

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Page 115 - Asymptotic expansions and analytic continuations for a class of Barnesintegrals

Page 115 - MORSE and H. FESHBACH. Methods of Theoretical Physics, Vol. I. McGraw Hill. New York, 1953.

Page 116 - AP Prudnikov, Yu.A. Brychkov and OI Marichev, Integrals and Series, Vol. 3 (Gordon & Breach, New York, 1990).