Movable singularities and quadrature

Abstract

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