# Alternative view of asymptotic expansions and the Stokes' phenomenon

*by Professor Ed Smith*

*Institution:*Department of Mathematics, La Trobe University

*Date: Fri 13th October 2006*

*Time: 3:15 PM*

*Location: Theatre 2, Ground Floor, 111 Barry Street, Carlton*

*Abstract*: The traditional view of asymptotic expansions, due to Stokes, is to use only the series out to its term of least magnitude. This is called "superasymptotics". Stokes also noted that solutions to second order linear differential equations had expansions which were linear combinations of expansions of two standard solutions of the equation, but that the coefficients were discontinuous across Stokes lines, behaviour known as Stokes phenomenon. Later (140 years) Berry and others used Borel summation of the (divergent) post superasymptotic series to derive expansions of solutions of differential equations which gave smooth variation of the multipliers according to a generic error function of scaled argument. In this seminar, I will consider asymptotic expansions of Laplace type integrals which are solutions of second order differential equations. They may be expanded in convergent series which embed the traditional asymptotic expansions. When the superasymptotic expansion (a finite series) is subtracted from this convergent expansion, a convergent series representation for the remainder is obtained. Stokes phenomenon then results from terms in this reminder series close to the cut off in the superasymptotic expansion, on both sides of the cut off. Stokes phenomenon is derived without resorting to divergent series.

** Drinks and nibbles will served after the seminar at the Australian Mathematical Sciences Institute (AMSI), Ground Floor, 111 Barry St (behind Theatre 2). **

*For More Information:* Emma Lockwood emmal@ms.unimelb.edu.au Tel: +61 3 8344 1617