A Fourier series method for numerically inverting the Laplace transform is described. The method has previously been analyzed only for real-valued functions. The method is extended to complex Fourier series and a complex-valued approximation. The analysis of the approximation is presented giving a bound for the error in discretization. Other types of errors are discussed and numerical results are presented. The sensitivity of the method to key parameters is documented and the error bound shown numerically to be very good. 10 refs., 5 figs.