A method for the numerical evaluation of finite integrals of oscillatory functions

However in physical problems the finiteness of the range of integration is often associated with a kind of natural boundary of f(x), such that it is impossible to extend f(x) to values of x beyond the upper limit b while preserving the general character of f(x). Analytically speaking, x = b may be a branch point of f(x). Alternatively, it may be possible to extend the range of integration to infinity as in equation (2), but the infinite integrals may not converge. As an example of the branch point difficulty we can consider the integral