An error estimate for Stenger’s quadrature formula

Abstract:

The basis of this paper is the quadrature formula \[ \int _{ - 1}^1 {f(x) dx \approx \log q\sum \limits _{m = - M}^M {\frac {{2{q^m}}}{{{{(1 + {q^m})}^2}}}f\left ( {\frac {{{q^m} - 1}}{{{q^m} + 1}}} \right )} ,} \] where $q = \exp (2h)$, h being a chosen step length. This formula has been derived from the Trapezoidal Rule formula by F. Stenger. An explicit form of the error is given for the case where the integrand has a factor of the form ${(1 - x)^\alpha }{(1 + x)^\beta },\alpha ,\beta > - 1$. Application is made to the evaluation of Cauchy principal value integrals with endpoint singularities and an appropriate error form is derived.