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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.
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