Complex-plane methods for evaluating highly oscillatory integrals in nuclear physics. II

Abstract

For pt.I see ibid., vol.14, p.973(1988). Products of Green functions for pion-nucleon interactions can most naturally be evaluated by transformation to a momentum-space representation. Such transformations often involve an integral whose integrand is highly oscillatory. By deforming the contour of integration in the complex plane, such an integral can be made to converge rapidly. All oscillatory terms are converted into pure damping terms which decrease exponentially. Examples are given for simple integrals, for the integral of three spherical Bessel functions, and for the Fourier and spherical Bessel function transforms of the Woods-Saxon potential (Fermi-Dirac distribution function). They authors also show that oscillatory principal-value integrals can usually be accurately evaluated using this method. Convergence with respect to the number of Gaussian points is studied. It is shown that the method normally converges much better than ordinary integration along the real axis, especially as the frequency of the oscillations increases relative to the damping. A comparison is made with the method of steepest descent.