Error bounds and Stokes' phenomenon in the theory of divergent expansions
By Prof. Vladimir P. Gurarii, Mathematics, Swinburne University of Technology
Thursday, August 20, 2009
Abstract
We consider classes of functions uniquely determined by coefficients of their divergent expansions. Approximating a function from such a class by partial sums of its expansion, we study how the accuracy changes when we move within a region of the complex plane. Analysis of these changes allows us to propose a theory of divergent expansions, which includes Stokes phenomenon as its essential part. This in turns enables us to formulate the necessary and sufficient conditions for an individual divergent expansion to encounter Stokes` phenomenon.
We show explicit expressions for exponentially small terms which appear upon crossing Stokes lines and which lead to improvement of accuracy. The main features of the theory will be illustrated using classical divergent expansions. Even for these well known expansions our approach reveals properties which have not been known previously:
* Euler factorial series 1/z-1!/z^2+2!/z^3-...;
* Stirling expansion for z!;
* Divergent expansions for Airy and Bessel functions.