Ramified Integrals, Singularities and Lacunas

ISBN-13: 9789401040952 / Angielski / Miękka / 2012 / 294 str.

V. a. Vassiliev

Opis

Many special functions occuring in physics and partial differential equations can be represented by integral transformatIons: the fundamental solutions of many PDE's, Newton-Coulomb potentials, hypergeometric functions, Feynman integrals, initial data of (inverse) tomography problems, etc. The general picture of such transfor mations is as follows. There is an analytic fibre bundle E --+ T, a differential form w on E, whose restrictions on the fibres are closed, and a family of cycles in these fibres, parametrized by the points of T and depending continuously on these points. Then the integral of the form w along these cycles is a function on the base. The analytic properties of such functions depend on the monodromy action, i.e., on the natural action of the fundamental group of the base in the homology of the fibre: this action on the integration cycles defines the ramification of the analytic continuation of our function. The study of this action (which is a purely topological problem) can answer questions about the analytic behaviour of the integral function, for instance, is this function single-valued or at least algebraic, what are the singular points of this function, and what is its asymptotics close to these points. In this book, we study such analytic properties of three famous classes of func tions: the volume functions, which appear in the Archimedes-Newton problem on in tegrable bodies; the Newton-Coulomb potentials, and the Green functions of hyperbolic equations (studied, in particular, in the Hada mard-Petrovskii-Atiyah-Bott-Garding lacuna theory)."

Spis treści

Introduction. I. Picard-Lefschetz-Pham theory and singularity theory. II. Newton's theorem on the nonintegrability of ovals. III. Newton's potential of algebraic layers. IV. Lacunas and the local Petrovskii condition for hyperbolic differential operators with constant coefficients. V. Calculation of local Petrovskii cycles and enumeration of local lacunas close to real function singularities. Appendix: a FORTRAN program searching for the lacunas and enumerating the morsifications of real function singularities. Bibliography. Index.