Theory and applications of dual asymptotic expansions

Abstract

This thesis introduces an original mathematical theory for a new kind of asymptotic expansion of real-analytic functions of two variables. The Dual Asymptotic Expansion (DAE) expresses a bivariate function asymptotically as a sum of products of univariate functions: the series is asymptotic in the univariate sense as each variable approaches its limiting value while the other variable remains fixed.

The DAE exists to infinitely many terms at almost every expansion point where the function is analytic; the set of exceptional points has Lebesgue measure zero. The terms of a DAE are uniquely determined by the choice of expansion point, and usually contain nonpolynomial functions. DAE's can approximate special functions by series of elementary functions with better accuracy than comparable Taylor or Pade approximations.

The thesis presents several applications and includes a small implementation of DAE methods in the MAPLY 5.4 computer algebra system.