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Ramified Integrals, Singularities and Lacunas (Hardcover, 1995 ed.)
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Ramified Integrals, Singularities and Lacunas (Hardcover, 1995 ed.)
Series: Mathematics and Its Applications, 315
Expected to ship within 12 - 17 working days
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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).
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