Quadrature rule for indefinite integral of algebraic–logarithmic singular integrands

An automatic quadrature method is presented for approximating the indefinite integral of functions having algebraic–logarithmic singularities Q ( x , y , c ; f ) = ∫ x y f ( t ) | t - c | α log | t - c | d t , - 1 ⩽ x , y , c ⩽ 1 , α > - 1 , within a finite range [ - 1 , 1 ] for some smooth function f ( t ) , that is approximated by a finite sum of Chebyshev polynomials. We expand the given indefinite integral in terms of Chebyshev polynomials by using auxiliary algebraic–logarithmic functions. Present scheme approximates the indefinite integral Q ( x , y , c ; f ) uniformly, namely bounds the approximation error independently of the value c as well x and y. This fact enables us to evaluate the integral transform Q ( x , y , c ; f ) with varied values of x, y and c efficiently. Some numerical examples illustrate the performance of the present quadrature scheme.