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Analytic continuation of solutions of integral equations and localization of singularities. (English. Russian original) Zbl 0897.45001
Differ. Equations 32, No. 11, 1541-1549 (1996); translation from Differ. Uravn. 32, No. 11, 1544-1553 (1996).
The paper is devoted to studying the singularities of the analytic continuation of solutions to Fredholm-type integral equations of the form \[ u(x)- \lambda\int_M K(x,y)u(y)dy= f(x) \] with sufficiently small complex parameter \(\lambda\), where \({\mathcal M}\) is a smooth manifold, \(x\in{\mathcal M}\), the kernel \(K(x,y)\) is an integrable function defined on the product \({\mathcal M}\times {\mathcal M}\). Under the assumption that \({\mathcal M}\) is the real part of some complex manifold \({\mathcal M}_c\), \(K(x,y)\) extends to be an analytic function on \({\mathcal M}_c\times {\mathcal M}_c\) with bounded singularities, and the function \(f(x)\) is extendable with respect to an analytic ramified function on \({\mathcal M}_c\), the author proves the existence of analytic continuations of solutions to the above equation which are analytic ramified functions with the set of bounded singularities. The proof is based on the representation of such solutions via Fredholm determinants.
45B05 Fredholm integral equations
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
30B40 Analytic continuation of functions of one complex variable