Some classes of singular integral equations solvable in a closed form
Let \(\Gamma\) be a simple smooth closed contour dividing the plane of the complex variable into the two domains \(D^+\ni 0\) and \(D^-\ni\infty\). The following singular integral equation is considered: \[ K\varphi(t) \equiv a_0(t)\varphi(t) + a_1(t)S[k_1\varphi](t) +\dots + a_n(t)S[k_n\varphi](t) =f(t),\tag{1} \] whose right-hand side is taken in the form \(f(t) = 2M_1(t)a_1(t) + \dots + 2M_n(t)a_n(t)\), where \(M_k(t)\), \(k =1,\dots, n\), are polynomials, \(S[\omega](t)\) stands for a singular operator, while \(a_0(t)\), \(a_i(t)\), \(k_i(t)\), \(i =1,\dots, n\), are \(H_\mu\)-continuous functions of the points of the contour. The author discusses the cases when the solution of equation (1) can be written in a closed form. An illustrative example is given.
singular integral equation with \(n\) kernels of Cauchy type
scientific article; zbMATH DE number 1956418