Summary
A method is proposed for the calculation of integrals\(I(b) = \int\limits_0^A {\frac{{g(x)dx}}{{(x^r + b^r )^s }}} \) and\(I\prime (b) = \int\limits_{ - A_1 }^{A_2 } {\frac{{g(x)dx}}{{(|x|^r + b^r )^s }}} \) for positive small values ofb withg(x) analytic in a disk centered at the origin andr positive rational. The integrals are splitted into a singular part and a regular function ofb r which can be extrapolated to smallb's. A principal value integral and special cases whereg(x) is not analytic at the origin are also considered.
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Midy, P., Yakovlev, Y. An extrapolation method for the quadrature of functions with singularities in the vicinity of the interval of integration. Numer. Math. 32, 183–196 (1979). https://doi.org/10.1007/BF01404874
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DOI: https://doi.org/10.1007/BF01404874