On the Numerical Evaluation of Singular Integrals of Cauchy Type

it is common practice to interpret (1 ) as an operator in the usual function spaces, e.g., in L,-spaces, p > 1. The function f(l) defined by (1) is often referred to as the finite Hilbert transform of f and many important and useful properties of such transformations are expressible in typical functiontheoretic language [8]. The abstract setting, however, is satisfactory only if function-theoretic properties of (1) are to be investigated but (as so very often happens when one turns to practical applications) the abstract results prove inadequate for direct numerical computations. Chaotic behavior of (1) of any sort is manifestly unacceptable and some type of uniform approximation theory is required. Such a theory does not seem to have been announced. Notwithstanding, the literature is replete with numerical methods for evaluating singular integrals of the form (1). Many of the methods have been motivated by problems in the field of aerodynamics (see, e.g., [1, 2, 11]) which require the solution of singular integral equations of the form