NUMERICAL QUADRATURE PERFORMED ON THE GENERALIZED PROLATE SPHEROIDAL FUNCTIONS

摘要

In the recent paper on the computation of the prolate spheroidal wave functions the numerical technique for their accurate integration was also proposed. The conventional quadrature formulas can not be used here: this would cause a significant lost of accuracy. The point is that the prolate spheroidal functions may vanish exponentially fast or oscillate rapidly, accumulating zeroes near singular points +-1. The necessity to compute integrals containing prolate spheroidal wave functions arises in many practical applications. They appear with the separation of variables in spheroidal coordinates as eigenfunctions of a singular Sturm-Lioville problem and constitute a natural basis in axially-symmetric physical problems. In this case integrals representing the Fourier coefficients or matrix elements are desired, which contain this functions or even their products.