The Shifted Square Wave

Consider the Fourier series of the unit-amplitude square wave, shifted along the $\theta$ axis to $\theta'$, with $\theta-\theta'= \pi/2$, so that it becomes an even function of $\theta$, as shown in Figure 5. The original function is given by the DP Fourier series

$$f_{\rm c}(\theta) = \frac{4}{\pi} \sum_{j=0}^{\infty} \frac{(-1)^{j}}{k}\, \cos(k\theta),$$

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

$$f_{\rm s}(\theta) = \frac{4}{\pi} \sum_{j=0}^{\infty} \frac{(-1)^{j}}{k}\, \sin(k\theta),$$

where $k=2j+1$. These two series are convergent almost everywhere, but not absolutely or uniformly convergent. There are two special points at $\theta=\pi/2$ and at $\theta=-\pi/2$, where we have for the original function $f_{\rm c}(\pi/2)=0$ and $f_{\rm c}(-\pi/2)=0$. At these points the original function is discontinuous and the corresponding FC function diverges logarithmically. The representation of the original function in terms of the first-order center series is given by

$$f_{\rm c}(\theta) = \frac{2}{\pi\cos(\theta)} \left\{ 1... ...fty} \frac{2(-1)^{j}}{k(k+2)}\, \cos[(k+1)\theta] \right\},$$

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

$$f_{\rm s}(\theta) = \frac{2}{\pi\cos(\theta)} \left\{ \... ...fty} \frac{2(-1)^{j}}{k(k+2)}\, \sin[(k+1)\theta] \right\},$$

where $k=2j+1$, for $\theta\neq\pi/2$ and $\theta\neq-\pi/2$. These two series are absolutely and uniformly convergent. The representation of the original function in terms of the second-order center series is given by

$$f_{\rm c}(\theta) = \frac{2}{3\pi\cos^{2}(\theta)} \left... ...\frac{12(-1)^{j}}{k(k+2)(k+4)}\, \cos[(k+2)\theta] \right\},$$

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

Figure 6: Parabolic wave: the original function (solid line) and its conjugate function (dashed line) plotted within the periodic interval $[-\pi ,\pi ]$. The dotted lines mark the special points.

$$f_{\rm s}(\theta) = \frac{2}{3\pi\cos^{2}(\theta)} \left... ...\frac{12(-1)^{j}}{k(k+2)(k+4)}\, \sin[(k+2)\theta] \right\},$$

where $k=2j+1$, for $\theta\neq\pi/2$ and $\theta\neq-\pi/2$. These two series are absolutely and uniformly convergent. The derivation of the center series can be found in Section A.5 of Appendix A, and the results of the tests are shown in the tables in Subsections B.9 and B.10 of Appendix B.