  In this paper we propose methods for computing Fresnel integrals based on
truncated trapezium rule approximations to integrals on the real line, these
trapezium rules modified to take into account poles of the integrand near the
real axis. Our starting point is a method for computation of the error function
of complex argument due to Matta and Reichel ({\em J. Math. Phys.} {\bf 34}
(1956), 298--307) and Hunter and Regan ({\em Math. Comp.} {\bf 26} (1972),
539--541). We construct approximations which we prove are exponentially
convergent as a function of $N$, the number of quadrature points, obtaining
explicit error bounds which show that accuracies of $10^{-15}$ uniformly on the
real line are achieved with N=12, this confirmed by computations. The
approximations we obtain are attractive, additionally, in that they maintain
small relative errors for small and large argument, are analytic on the real
axis (echoing the analyticity of the Fresnel integrals), and are
straightforward to implement.
