A method for indefinite integration of oscillatory and singular functions

摘要

We propose a general method for computing indefinite integrals of the form I( y) = integral(y)(0) y g(t)k(t)dt (0 <= y <= y(max)), where g is a smooth function, and k is a function that contains a singular factor or is rapidly oscillatory. The only assumption on k is that it satisfies a linear differential equation with polynomial coefficients. The approximate value of the integral is given in terms of Chebyshev coefficients of functions that form a solution of a certain system of differential equations. As an illustration, we present effective algorithms for computing indefinite integrals of the functions g(t)|t - d|(alpha)e(iwt), g(t) log |t - d| e(iwt), g(t) t(alpha) J(v)(ct).