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In this paper we demonstrate that the numerical method of steepest descentfails when applied in a straight forward fashion to the most commonly occurringhighly oscillatory integrals in scattering theory. Through a polar change ofvariables, however, the integral can be brought on a form that can be solvedefficiently using a mix of oscillatory integration techniques and classicalquadrature. The approach is described in detail and demonstrated numerically onintegration problems taken from applications.

Applying the numerical method of steepest descent on multivariate oscillatory integrals in scattering theory