  This article is concerned with a new method for the approximate evaluation of
Fourier sine and cosine transforms. We develop and analyse a new quadrature
rule for Fourier sine and cosine transforms involving transforming the integral
to one over the entire real line and then using the trapezoidal rule in order
to approximate the transformed integral. This method follows on from the work
of Ooura and Mori.
  A complete error analysis is made using contour integration. An example is
examined in detail and the error is analysed using residues and the saddle
point method. The method we have developed is characterised by its simplicity
and single exponential convergence.
