Error analysis for a sinhsinh transformation used in evaluating nearly singular boundary element integrals

In the two-dimensional boundary element method, one often needs to evaluate numerically integrals of the form ∫-11g(x)j(x)f((x-a)2+b2)dx where j2j2 is a quadratic, g is a polynomial and f is a rational, logarithmic or algebraic function with a singularity at zero. The constants a and b   are such that -1⩽a⩽1-1⩽a⩽1 and 0