Regularized Computation of Oscillatory Integrals with Stationary Points

Ability to calculate integrals of rapidly oscillating functions is crucial for solving many problems in optics, electrodynamics, quantum mechanics, nuclear physics, and many other areas. The article considers the method of computing oscillatory integrals using the transition to the numerical solution of the system of ordinary differential equations. Using the Levin’s collocation method, we reduce the problem to solving a system of linear algebraic equations. In the case where the phase function has stationary points, (its derivative vanishes on the interval of integration) the solution of the corresponding system becomes an ill-posed task. The regularized algorithm presented in the article describes the stable method of integration of rapidly oscillating functions at the presence of stationary points. Performance and high accuracy of the algorithms are illustrated by various examples.