A Trigonometric Quadrature Rule for Cauchy Integrals with Jacobi Weight

In this paper, we consider a quadrature rule for Cauchy integrals of the form I(wf;s)=[formula]w(t)f(t)/(t-s)dt, -1<s<1, for a smooth density function f(t) and Jacobi's weights w(t)=(1-t)^α(1+t)^β, α, β>-1/2. Using the change of variables t=cosy, s=cosx and subtracting out the singularity, we propose a trigonometric quadrature rule. We obtain the error bounds independent of the set of values of poles and construct an automatic quadrature of nonadaptive type.