Uniform approximation to Cauchy principal value integrals with logarithmic singularity

Abstract

An approximation of Clenshaw–Curtis type is given for Cauchy principal value integrals of logarithmically singular functions I(f;c)=−∫−11f(x) (log|x−c|)∕(x−c)dx (c∈(−1,1)) with a given function f. Using a polynomial pN of degree N interpolating f at the Chebyshev nodes we obtain an approximation I(pN;c)≅I(f;c). We expand pN in terms of Chebyshev polynomials with O(NlogN) computations by using the fast Fourier transform. Our method is efficient for smooth functions f, for which pN converges to f fast as N grows, and so simple to implement. This is achieved by exploiting three-term inhomogeneous recurrence relations in three stages to evaluate I(pN;c). For f(z) analytic on the interval [−1,1] in the complex plane z, the error of the approximation I(pN;c) is shown to be bounded uniformly. Using numerical examples we demonstrate the performance of the present method.