Derivations of Center Series

In this appendix we give short but complete derivations of all the center series used in this paper. In each case we start with the complex power series $S_{z}$ related to the original DP Fourier series, and construct from it the complex center series $C_{z}$. This involves the positions of the singularities of the corresponding inner analytic function $w(z)$ on the unit circle. We then write $S_{z}$ in terms of $C_{z}$ and take the real and imaginary parts, in order to obtain the center series corresponding to the original DP Fourier series and of its FC series.

One must keep in mind that the final forms obtained for the functions $f_{\rm c}(\theta)$ and $f_{\rm s}(\theta)$ in terms of the center series are only valid away from the special points on the periodic interval, which correspond to the singular points of $w(z)$ on the complex plane. The values of the DP real functions at these special points are usually determined very easily by the original Fourier series.

Starting from eight Fourier Conjugate pairs of DP Fourier series, we derive eight pairs of first-order center series and six pairs of second-order center series. Of these 44 series a total of 40 series were used in this paper, of which 16 DP Fourier series and 16 first-order center series were tested against each other.