A Singular Cosine Series

Consider the Fourier series of the Dirac delta ``function'' centered at $\theta=\theta_{1}$, which we denote by $\delta(\theta-\theta_{1})$. We may easily calculate its Fourier coefficients using the rules of manipulation of $\delta(\theta-\theta_{1})$, thus obtaining $\alpha_{k}=\cos(k\theta_{1})/\pi$ and $\beta_{k}=\sin(k\theta_{1})/\pi$ for all $k$. The series is therefore the complete Fourier series given by

$$\begin{eqnarray*} S_{\rm c} & = & \frac{1}{2\pi} + \frac{1}{\pi} \sum_{k=1... ...pi} + \frac{1}{\pi} \sum_{k=1}^{\infty} \cos(k\Delta\theta), \end{eqnarray*}$$

where $\Delta\theta=\theta-\theta_{1}$. Apart from the constant term this is in fact a cosine series on this new variable. Clearly, this series diverges at all points in the interval $[-\pi,\pi]$. Undaunted by this, we construct the FC series, with respect to the new variable $\Delta\theta$,

$$\bar{S}_{\rm c} = \frac{1}{\pi} \sum_{k=1}^{\infty} \sin(k\Delta\theta),$$

a series that is also divergent, this time almost everywhere. If we define $v=\exp(\mbox{\boldmath$\imath$}\theta)$ and $v_{1}=\exp(\mbox{\boldmath$\imath$}\theta_{1})$ the corresponding complex series $S_{v}$ is then given by

$$S_{v} = \frac{1}{2\pi} + \frac{1}{\pi} \sum_{k=1}^{\infty} \left(\frac{v}{v_{1}}\right)^{k},$$

where we included the $k=0$ term, and the corresponding complex power series $S_{z}$ is given by

$$S_{z} = \frac{1}{2\pi} + \frac{1}{\pi} \sum_{k=1}^{\infty} \left(\frac{z}{z_{1}}\right)^{k},$$

where $z=\rho v$ and $z_{1}=v_{1}$ is a point on the unit circle. This is a geometrical series that converges everywhere strictly inside the unit disk, where it therefore converges to the analytic function

$$w_{\delta}(z) = \frac{1}{2\pi} - \frac{1}{\pi}\, \frac{z}{z-z_{1}}.$$

This function has a simple pole at the singular point of the Dirac delta ``function''. This is a hard singularity of degree $1$. As was discussed in Section 6, strictly speaking this is not an inner analytic function by the definition which we adopted here, for two reasons. First, $w_{\delta}(0)$ is not zero, which is easily fixed by just taking off the constant term. Second, it does not reduce to a real function over the real axis. However, as was shown in Section 6, it does reduce to a real function over the straight line $z=\chi z_{1}$, with real $\chi$, since in this case we have

$$w(z) = -\, \frac{1}{\pi}\, \frac{\chi}{\chi-1},$$

which is a real function of the real variable $\chi$. Since we have $z_{1}=\exp(\mbox{\boldmath$\imath$}\theta_{1})$, we see that this is an inner analytic function rotated by an angle $\theta_{1}$ around the origin. We see therefore that our definition of inner analytic function can be easily generalized in this way, to make reference to any straight line going through the origin. We also see that there is nothing to prevent us from treating this function just like any other inner analytic function.

We now examine the real part of the function $w_{\delta}(z)$, which was derived in Section 6, and which is related to the series $S_{\rm c}$,

$$\Re[w_{\delta}(z)] = \frac{1}{2\pi} - \frac{1}{\pi}\, ... ...a)\right]} {\left(\rho^{2}+1\right)-2\rho\cos(\Delta\theta)}.$$

The $\rho\to 1$ limit of $\Re[w_{\delta}(z)]$ gives us back the constant term and the original cosine series $S_{\rm c}$, and in fact attributes well-defined finite values to it almost everywhere, even though the original series is divergent. As was shown directly in Section 6, in the $\rho\to 1$ limit this real part has all the required properties of the Dirac delta ``function'', including the fact that it assumes the value zero almost everywhere. Note that although the singularity of $w_{\delta}(z)$ at $z=z_{1}$ is a simple pole and thus not an integrable one, the real part of $w_{\delta}(z)$ is integrable if one crosses the singularity in a specific direction, which in this case is the direction of the integration along the unit circle.

It is interesting that we may also write a closed expression for the real function which is the FC function of the Dirac delta ``function''. We just consider the imaginary part of $w_{\delta}(z)$, which is related to the series $\bar{S}_{\rm c}$,

$$\Im[w_{\delta}(z)] = \frac{1}{\pi}\, \frac {\rho\sin(\Delta\theta)} {\left(\rho^{2}+1\right)-2\rho\cos(\Delta\theta)}.$$

We now take the $\rho\to 1$ limit, assuming that $\Delta\theta\neq 0$, and since this is an actual function we may write

$$\begin{eqnarray*} \bar{f}_{\rm c}(\Delta\theta) & = & \lim_{\rho\to 1} \Im[w... ...rac{1}{2\pi}\, \frac{1+\cos(\Delta\theta)}{\sin(\Delta\theta)}. \end{eqnarray*}$$

This is just a regular function except for a simple pole at $\Delta\theta=0$, and in fact quite a simple rational function involving trigonometric functions.

Note that we get a free set of integration formulas out of this effort. Since the coefficients $a_{k}$ can be written as integrals involving $\bar{f}_{\rm c}(\Delta\theta)$, we have at once that for all $k>0$

$$\int_{-\pi}^{\pi}d(\Delta\theta)\, \frac{1+\cos(\Delta\theta)}{\sin(\Delta\theta)}\, \sin(k\Delta\theta) = 2\pi.$$

We may easily construct from this single Dirac delta ``function'' a pair of such ``functions'' which relates to the derivative of the standard square wave. All we have to do is to use the values $\theta_{1}=0$ and $\theta_{1}=\pi$ and add to the delta ``function'' at zero given by $2\delta(\theta-0)$ the delta ``function'' at $\pi$ given by $-2\delta(\theta-\pi)$. If we call the corresponding analytic functions $w_{0}(z)$ and $w_{\pi}(z)$, we have for the function $w_{2\delta}(z)$ corresponding to the pair

$$\begin{eqnarray*} w_{0}(z) & = & \frac{1}{\pi} + \frac{2}{\pi}\, \frac{z}{... ... \frac{z}{1+z} \\ & = & \frac{4}{\pi}\, \frac{z}{1-z^{2}}. \end{eqnarray*}$$

This is an inner analytic function with two simple poles, one at $z=1$ and one at $z=-1$. If, on the other hand, we consider the logarithmic derivative of the analytic function $w_{\rm sq}(z)$ of the standard square wave, we get

$$\begin{eqnarray*} z\, \frac{dw_{\rm sq}(z)}{dz} & = & \frac{2}{\pi}\, z\, ... ...}{1-z} \right) \\ & = & \frac{4}{\pi}\, \frac{z}{1-z^{2}}. \end{eqnarray*}$$

We see therefore that we do indeed have

$$w_{2\delta}(z) = z\, \frac{dw_{\rm sq}(z)}{dz}.$$

In terms of derivatives with respect to $\theta$ this is written as

$$w_{2\delta}(z) = -\mbox{\boldmath$\imath$}\, \frac{dw_{\rm sq}(z)}{d\theta},$$

where once again the factor of $-\mbox{\boldmath$\imath$}$ has the role of exchanging real and imaginary parts, and thus sines and cosines.

Observe that this is an example in which, although the function $w_{\delta}(z)$ is analytic on the open unit disk, the series $S_{z}$ is not convergent on any points of the unit circle, and hence the two FC Fourier series do not converge together on any points of the unit circle. It is therefore outside the hypotheses we used in most of this paper, but it is still full of meaning. This indicates that there is more in this structure than has been analyzed so far. If fact, it is quite possible that the whole structure of distributions such as the one related to the Dirac delta ``function'' is lurking on the rim of the maximum disk of convergence of the series $S_{z}$ of the inner analytic functions $w_{\delta}(z)$ corresponding to divergent trigonometric series, that is, on the rim of the unit disk.