Singular equal equations

The classical analytic solution of the dominant singular integral equation ... is found by transforming the equation into a Riemann boundary problem. (The above integral is interpreted as a Cauchy principal value.) This analytic solution is not very useful for numerical work, since it requires the evaluation of a Cauchy principal value integral of a function Z, which has, in general, algebraic logarithmic singularities at ±1. Three alternative solutions are found which avoid this problem. These solutions are used as the basis of an algorithm for the numerical solution of the above singular integral equation. Convergence of this algorithm is proved, and four examples are given, including the H equation of Chandrasekhar. An algorithm and a computer program is also given for the solution of the complete singular integral equation ... Two examples are given, including the numerical solution of a singular integral equation arising in neutron transport theory.