  The capacitance of the circular parallel plate capacitor is calculated by
expanding the solution to the Love integral equation into a Fourier cosine
series. Previously, this kind of expansion has been carried out numerically,
resulting in accuracy problems at small plate separations. We show that this
bottleneck can be alleviated, by calculating all expansion integrals
analytically in terms of the Sine and Cosine integrals. Hence, we can, in the
approximation of the kernel, use considerably larger matrices, resulting in
improved numerical accuracy for the capacitance. In order to improve the
accuracy at the smallest separations, we develop a heuristic extrapolation
scheme that takes into account the convergence properties of the algorithm. Our
results are compared with other numerical results from the literature and with
the Kirchhoff result. Error estimates are presented, from which we conclude
that our results is a substantial improvement compared with earlier numerical
results.
