On the solution of singular integral equations with automorphic kernels by mechanical quadrature (Q1912440)

Consider a singular integral equation with automorphic kernel \[ A(x)\varphi(x)+ B(x) \int_\Gamma \varphi(\tau)\Biggl[{1\over \tau- x}+ \sum^{m- 1}_{j= 1} \Biggl({1\over \tau- \sigma_j(x)}- {1\over \tau- \sigma_j(\infty)}\Biggr)\Biggr] d\tau= f(x),\;x\in \Gamma.\tag{1} \] Let the coefficients \(A(x)\) and \(B(x)\) of the equation and its right-hand side \(f(x)\) satisfy a Hölder condition with exponent \(\lambda\) at the interior points of the line \(\Gamma\), let \(A^2(x)- \pi^2 B^2(x)\neq 0\) everywhere on \(\Gamma\), and let the line \(\Gamma\) be contained in the interior of one of the fundamental domains of the group. For simplicity of reasoning we shall assume that the fundamental domain is symmetric with respect to the real axis. The kernel of equation (1) is the automorphic analog of the Cauchy kernel and hence in terms of its solvability equation (1) is an analog of a singular integral equation with the Cauchy kernel. Quadrature formulas are obtained for singular integral equations with automorphic analogs of the Cauchy kernel and an algorithm is developed for solving equation (1) by mechanical quadrature in the case when \(\Gamma= [\alpha, \beta]\) is an interval of the real axis.