Appendix: Examples of Center Series

In this appendix we will give a few simple illustrative examples of the construction of center series. In many cases the construction of a center series constitutes a practical way to determine the corresponding real function, and to thus take advantage of the very existence of the inner analytic function, for example when it is not possible to exhibit that function in closed form, in order to explicitly take its limit to the unit circle.

The process of construction consists of three parts, starting with the determination of the power series $S_{z}$ from the original DP Fourier series, which is simple and can always be done without any difficulty. The second step is the construction of the complex center series $C_{z}$, which is operationally fairly simple but depends on the knowledge of the complete set of dominant singularities over the unit circle of the inner analytic function $w(z)$ that the series $S_{z}$ converges to.

The last step is the recovery from $S_{z}$ written in terms of $C_{z}$ of the real and imaginary parts of $w(z)$, in order to obtain the center series versions of the original DP Fourier series and of its FC series. This is straightforward but can become, in some cases, a rather long algebraic process. In each example we will develop explicitly all these steps, with a reasonable amount of detail.

Some of the examples that follow are the same that were worked out by another method in the appendices of the already mentioned previous paper [1]. They are presented in the same order as in that paper. It is understood that the final forms obtained for the functions $f_{\rm s}(\theta)$ and $f_{\rm c}(\theta)$ in terms of the center series are valid only away from the special points.