Movable singularities and quadrature

Goodrich, R. F.; Stenger, F.

doi:10.1090/S0025-5718-1970-0275669-4

Movable singularities and quadrature
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by R. F. Goodrich and F. Stenger PDF

Math. Comp. 24 (1970), 283-300 Request permission

Abstract:

A general procedure is described for treating a movable singularity in an integral. This enabĺes us to change the original integral ${I_0}$ into $G{I_1}$, where $G$ depends only on the parameters of the singularity and ${I_1}$, is a new integral which exists for all values of the parameters. The results are then applied to the particular problem of evaluating \[ \int _{ - 1}^1 {\frac {{f(x)dx}} {{{{\{ (1 - {x^2})(1 - {k^2}{x^2})\} }^{1/2}}}}} ,\] where $f$ is entire and $k$ varies between $0$ and $1$. Some new quadrature schemes and new effective methods of evaluating incomplete elliptic integrals are derived.

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