Borel-Laplace transform and asymptotic theory. Introduction to resurgent analysis. (English) Zbl 0852.34001

Sternin, Boris Yu.; Shatalov, Victor

Boca Raton, FL: CRC Press. 270 p. (1996).

Methods and ideas of resurgent analysis are discussed. By examples from the asymptotic theory of differential equations it is shown how the main notions of resurgent analysis arise while investigating solutions to differential equations. Singular points of ordinary differential equations, equations on an infinite cylinder, and semi-classical approximations are dealt with. In Chapter I the Borel-Laplace transform of ramifying analytic functions is studied to construct the “spectral analysis” of resurgent functions theory. The notions of analytic hyperfunction, microfunction, and generalized hyperfunction, necessary in the resurgent function theory, are introduced.
In Chapter II resurgent analysis itself is considered. The main properties of the notion of resurgent functions of several independent variables are investigated. Near focal points and the connection homomorphs are discussed. Examples are given.
In Chapter III applications are analyzed for illustration. They relate to ordinary and partial differential equations. The saddle point method is used for Laplace integral study by resurgent analysis. In the Appendix integral transforms of ramifying analytic functions are discussed.

Reviewer: V.Burjan (Praha)

MSC:

34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations
44A10	Laplace transform
35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
34M99	Ordinary differential equations in the complex domain
34E20	Singular perturbations, turning point theory, WKB methods for ordinary differential equations

Keywords:

spectral analysis; resurgent analysis; asymptotic theory of differential equations; Borel-Laplace transform