This thesis is concerned with the efficient (accurate and fast) computation, via modified trapezium rules, of some special functions which can be written as integrals of the form l~f(t) dt, where f (t) = e-pr2F(t), p > 0, and F is an even meromorphic function with simple poles in a strip surrounding the real line. Specifically, this thesis considers the approximation of the Fresnel integrals, the complementary error function of complex argument and the Faddeeva function, and the 2D impedance half-space Green's function for the Helmholtz equation. The trapezium rule is exponentially convergent when F is analytic in a strip surrounding the real axis. In the case of meromorphic functions with simple poles, the trapezium rule can be modified to take into account the presence of these poles. The effect of truncating this modified trapezium rule is considered and specific approximations with explicit choices for step-size and number of quadrature points are given. Rigourous bounds for the errors are proven using complex analysis methods, and numerical calculations that demonstrate the accuracy of these approximations compared with the best known methods are also provided.