Series representations of Fourier integrals

General series representations, valid at least for small values of $x$, are obtained for the representative Fourier integral $\int _0^\infty {A\left ( k \right ) \exp \left ( {ikx} \right ) dk}$ for a variety of asymptotic forms of behaviour of the function $A\left ( k \right )$, assumed bounded and integrable in any finite range. The results obtained should be of value in the numerical evaluation of such integrals as well as in the determination of their analytic properties.