Fourier Transforms and the Method of Stationary Phase

Some theorems are derived for the asymptotic behaviour of the Fourier transform of a generalized function under conditions which are likely to occur in applied mathematics and which should be capable of relatively straightforward verification. These theorems are then applied to integrals of the type $∫-∞∞g(x)e-iaf(x)dx$ when the derivative of f is non-vanishing. Further theorems are developed to cover the case when the derivative of f vanishes at a point. In this way the validity of the method of stationary phase, together with a useful estimate of the error, is established under circumstances of practical importance.