On the solution of integral equations of the first kind with singular kernels of Cauchy-type

Abstract Two efficient quadrature formulae have been developed for evaluating numerically certain singular integral equations of the first kind over the finite interval [-1,1]. Central to this work is the application of four special cases of the Jacobi polynomials P n (x), whose zeros served as interpolation and collocation nodes: (i) α = β = −2 , Tn(x), the first kind Chebyshev polynomials (ii) α = β = 12 ,Un(x), the second kind (iii) α = −2 , β = 12 ,Vn(x), the third kind (iv) α = 12 , β = −2 ,Wn(x), the fourth kind. Four tables of numerical results have been provided for verification and validation of the rules developed.