The Parabolic Wave

Consider the Fourier series of a unit-amplitude periodic function built with segments of quadratic functions, joined together so that the resulting function is continuous and differentiable, in such a way that the result is an odd function of $\theta$, as shown in Figure 6. The original function is given by the DP Fourier series

$$f_{\rm s}(\theta) = \frac{32}{\pi^{3}} \sum_{j=0}^{\infty} \frac{1}{k^{3}}\, \sin(k\theta),$$

where $k=2j+1$, and the corresponding FC function is given by the DP Fourier series

$$f_{\rm c}(\theta) = \frac{32}{\pi^{3}} \sum_{j=0}^{\infty} \frac{1}{k^{3}}\, \cos(k\theta),$$

Figure 7: First slow-converging case: the original function (solid line) and its conjugate function (dashed line) plotted within the periodic interval $[-\pi ,\pi ]$. The dotted line marks the special point.

where $k=2j+1$. These two series are absolutely and uniformly convergent. There are two special points at $\theta=0$ and at $\theta=\pm\pi$, where we have for the original function $f_{\rm s}(0)=0$ and $f_{\rm s}(\pm\pi)=0$. At these points both the original function and the corresponding FC function have singularities on their second derivatives. The representation of the original function in terms of the first-order center series is given by

$$f_{\rm s}(\theta) = \frac{16}{\pi^{3}\sin(\theta)} \left... ...rac{6k(k+2)+8}{k^{3}(k+2)^{3}}\, \cos[(k+1)\theta] \right\},$$

where $k=2j+1$, for $\theta\neq 0$ and $\theta\neq\pm\pi$, and the representation of the corresponding FC function in terms of the first-order center series is given by

$$f_{\rm c}(\theta) = \frac{16}{\pi^{3}\sin(\theta)} \left... ...rac{6k(k+2)+8}{k^{3}(k+2)^{3}}\, \sin[(k+1)\theta] \right\},$$

where $k=2j+1$, for $\theta\neq 0$ and $\theta\neq\pm\pi$. These two series are absolutely and uniformly convergent. The derivation of the center series can be found in Section A.6 of Appendix A, and the results of the tests are shown in the tables in Subsections B.11 and B.12 of Appendix B.