The Riemann–Hilbert problem and the generalized Neumann kernel

On eigenfunctions of the integral operator with Neumann kernel

“…The function N(·, ·) is called the generalized Neumann kernel formed with A and η [6]. For A = 1 it reduces to the well-known Neumann kernel.…”

“…Application of Sokhotcky formula and straightforward calculations give the following lemma [6] Lemma 2…”

“…Using the properties of Cauchy's integral formula and Sokhotcky formula was proven the following lemma in [6] …”

“…The Riemann-Hilbert (RH) problems on simply connected regions [6] and multiply connected regions [7] are reduced to Fredholm integral equations of the second kind with a generalized Neumann kernel. It has been shown that the problem of conformal mapping, Dirichlet problem, Neumann problem and mixed Dirichlet-Neumann problem can all be treated as RH problems and solved efficiently using integral equations with the generalized Neumann kernel [1,3,4,5].…”

“…Note that if κ ≤ 0 ( κ > 0), then the number 1 ( −1 ) is an eigenvalue with multiplicity n(1) = max{0, −2k + 1} (n(−1) = max{0, 2k − 1}) [6]. In the paper we find all of eigenfunctions of N corresponding to the number ±1 and using the properties of eigenfunctions we solve Riemann-Hilbert boundary value problems, and give a method of construction of some class of conformal mappings.…”