Krein's method with certain singular kernel for solving the integral equation of the first kind (Q1337020)

For the integral operator with the logarithmic kernel \[ K_ \alpha \varphi (x) = \int_{-1}^ 1 \left[ \ln {1 \over | x - y |} + d \right] \varphi (y) dy \] the authors prove that the equalities \[ K_ dT^ 0_{2n} = \begin{cases} {\pi \over 2n} T_{2n} (x), & n = 1,2, \dots \\ {\pi \over \ln 2 + d}, & n = 0, \end{cases} \] \[ K_ dT_{2n-1} = {\pi \over 2n-1} T_{2n-1}, \quad n = 1,2, \dots, \] where \(T_ j\) denote the Chebyshev polynomials and \(T^ 0_ j(x) = (1-x^ 2)^{-1/2} T_ j(x)\).