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Goodwin, E.T. (1949) The Evaluation of Integrals of the Form ∫_(-∞)^∞?f(x)e^-x2dx. Proceedings of the Cambridge Philosophical Society, 45, 241-245.
https://doi.org/10.1017/S0305004100024786

has been cited by the following article:

• TITLE: Quadrature Rules for Functions with a Mid-Point Logarithmic Singularity in the Boundary Element Method Based on the x = tp Substitution

  AUTHORS: Stephen M. Kirkup, Javad Yazdani, George Papazafeiropoulos

  KEYWORDS: Boundary Element Method, Singular Integral, Numerical Integration

  JOURNAL NAME: American Journal of Computational Mathematics, Vol.9 No.4, December 16, 2019

  ABSTRACT: Quadrature rules for evaluating singular integrals that typically occur in the boundary element method (BEM) for two-dimensional and axisymmetric three-dimensional problems are considered. This paper focuses on the numerical integration of the functions on the standard domain [-1, 1], with a logarithmic singularity at the centre. The substitution x = tp, where p (≥ 3) is an odd integer is given particular attention, as this returns a regular integral and the domain unchanged. Gauss-Legendre quadrature rules are applied to the transformed integrals for a number of values of p. It is shown that a high value for p typically gives more accurate results.