Improvement of the asymptotic behaviour of the Euler-Maclaurin formula for Cauchy principal value and Hadamard finite-part integrals

In the recent works (Commun. Numer. Meth. Engng 2001; 17:881; to appear), the superiority of the non-linear transformations containing a real parameter b = 0 has been demonstrated in numerical evaluation of weakly singular integrals. Based on these transformations, we define a so-called parametric sigmoidal transformation and employ it to evaluate the Cauchy principal value and Hadamard finitepart integrals by using the Euler–Maclaurin formula. Better approximation is expected due to the prominent properties of the parametric sigmoidal transformation of whose local behaviour near x = 0 is governed by parameter b. Through the asymptotic error analysis of the Euler–Maclaurin formula using the parametric sigmoidal transformation, we can observe that it provides a distinct improvement on its predecessors using traditional sigmoidal transformations. Numerical results of some examples show the availability of the present method