Description
This book presents an extensive overview of logarithmic integral operators with kernels depending on one or several complex parameters. Solvability of corresponding boundary value problems and determination of characteristic numbers are analyzed by considering these operators as operator-value functions of appropriate complex (spectral) parameters. Therefore, the method serves as a useful addition to classical approaches. Special attention is given to the analysis of finite-meromorphic operator-valued functions, and explicit formulas for some inverse operators and characteristic numbers are developed, as well as the perturbation technique for the approximate solution of logarithmic integral equations. All essential properties of the generalized single- and double-layer potentials with logarithmic kernels and Green’s potentials are considered. Fundamentals of the theory of infinite-matrix summation operators and operator-valued functions are presented, including applications to the solution of logarithmic integral equations. Many boundary value problems for the two-dimensional Helmholtz equation are discussed and explicit formulas for Green’s function of canonical domains with separated logarithmic singularities are presented.
