Efficient computation of highly oscillatory Fourier-type integrals with monomial phase functions and Jacobi-type singularities.

We present a method for fast computation of highly oscillatory Fourier-type integrals with monomial phase functions and Jacobi-type singularities. It is shown that on an appropriately chosen contour in the complex plane the original integral can be expressed as the sum of two nonoscillatory and rapidly decaying integrals, each of which can be efficiently computed by the generalized Gauss-Laguerre rule or by the double exponential method. For moderate and large frequencies, the proposed method is stable and more efficient than the existing methods. We also give a Mathematica program for the implementation of the method on a computer. The effectiveness and accuracy is tested by means of numerical examples. [ABSTRACT FROM AUTHOR]