Classification of Singularities

Let us establish a general classification of the singularities of inner analytic functions on the unit circle. While it is possible that the scheme of classification that we describe here may have its uses in more general settings, for definiteness we consider here only the case of inner analytic functions that have one or more singularities on the unit circle. We also limit our attention to only those singular points, and ignore any singularities that may exist strictly outside the closed unit disk. In order to do this we must first establish some preliminary facts.

To start with, let us show that the operation of logarithmic differentiation stays within the set of inner analytic functions. Let us recall that, if $w(z)$ is an inner analytic function, then it has the properties that it is analytic on the open unit disk, that it is the analytic continuation of a real function on the real interval $(-1,1)$, and that $w(0)=0$. These last two are consequences of the fact that its Taylor series has the form

$$w(z) = \sum_{k=1}^{\infty} a_{k}z^{k},$$

with real $a_{k}$. We define the logarithmic derivative of $w(z)$ as

$$w^{\mbox{\Large$\cdot$}\!}(z) = z\, \frac{dw(z)}{dz},$$

which also establishes our notation for it. We might also write the first logarithmic derivative as $w^{1\!\mbox{\Large$\cdot$}\!}(z)$. Let us show that the logarithmic derivative of an inner analytic function is another inner analytic function. First, since the derivative of an analytic function is analytic in exactly the same domain as that function, and since the identity function $z$ is analytic in the whole complex plane, it follows that if $w(z)$ is analytic on the open unit disk, then so is $w^{\mbox{\Large$\cdot$}\!}(z)$. Second, if we calculate $w^{\mbox{\Large$\cdot$}\!}(z)$ using the series representation of $w(z)$, which converges in the open unit disk, since $w(z)$ is analytic there, we get

$$w^{\mbox{\Large$\cdot$}\!}(z) = z \sum_{k=1}^{\infty} ka_{k}z^{k-1},$$

since a convergent power series can always be differentiated term-by-term. It follows that, if the coefficients $a_{k}$ are real, then so are the new coefficients $ka_{k}$, so that the coefficients of the Taylor series of $w^{\mbox{\Large$\cdot$}\!}(z)$ are real, and hence it too is the analytic continuation of a real function on the real interval $(-1,1)$. Lastly, due to the extra factor of $z$ we have that $w^{\mbox{\Large$\cdot$}\!}(0)=0$. Hence, the logarithmic derivative $w^{\mbox{\Large$\cdot$}\!}(z)$ is an inner analytic function.

Next, let us define the concept of logarithmic integration. This is the inverse operation to logarithmic differentiation, with the understanding that we always choose the value zero for the integration constant. The logarithmic primitive of $w(z)$ may be defined as

$$w^{-1\!\mbox{\Large$\cdot$}\!}(z) = \int_{0}^{z}dz'\, \frac{1}{z'}\, w(z').$$

Let us now show that the operation of logarithmic integration also stays within the set of inner analytic functions. Note that since $w(0)=0$ the integrand is in fact analytic on the open unit disk, if we define it at $z=0$ by continuity, and therefore the path of integration from $0$ to $z$ is irrelevant, so long as it is contained within that disk. It is easier to see this using the series representation of $w(z)$, which converges in the open unit disk, since $w(z)$ is analytic there,

$$w^{-1\!\mbox{\Large$\cdot$}\!}(z) = \int_{0}^{z}dz'\, \sum_{k=1}^{\infty} a_{k}z^{\prime(k-1)}.$$

It is clear now that the integrand is a power series which converges within the open unit disk, and thus converges to an analytic function there. Since the primitive of an analytic function is analytic in exactly the same domain as that function, it follows that the logarithmic primitive $w^{-1\!\mbox{\Large$\cdot$}\!}(z)$ is analytic on the open unit disk. If we now execute the integration using the series representation, we get

$$w^{-1\!\mbox{\Large$\cdot$}\!}(z) = \sum_{k=1}^{\infty} \frac{a_{k}}{k}\, z^{k},$$

since a convergent power series can always be integrated term-by-term. Since the coefficients $a_{k}$ are real, so are the new coefficients $a_{k}/k$, and therefore $w^{-1\!\mbox{\Large$\cdot$}\!}(z)$ is the analytic continuation of a real function on the real interval $(-1,1)$. Besides, one can see explicitly that $w^{-1\!\mbox{\Large$\cdot$}\!}(0)=0$. Therefore, $w^{-1\!\mbox{\Large$\cdot$}\!}(z)$ is an inner analytic function. It is now easy to see also that the logarithmic derivative of this primitive gives us back $w(z)$.

Finally, let us show that the derivative of $w(z)$ with respect to $\theta$ is given by the logarithmic derivative of $w(z)$. Since we have that $z=\rho\exp(\mbox{\boldmath$\imath$}\theta)$, we may at once write that

$$\begin{eqnarray*} \frac{dw(z)}{d\theta} & = & \frac{dz}{d\theta}\, \frac{dw(... ... & = & \mbox{\boldmath$\imath$}w^{\mbox{\Large$\cdot$}\!}(z). \end{eqnarray*}$$

In the limit $\rho\to 1$, where and when that limit exists, the derivative of $w(z)$ with respect to $\theta$ becomes the derivatives of the limiting functions $f_{\rm c}(\theta)$ and $f_{\rm s}(\theta)$ of the FC pair of DP Fourier series. The factor of $\mbox{\boldmath$\imath$}$ effects the interchange of real and imaginary parts, and the change of sign, that are consequences of the differentiation of the trigonometric functions.

We see therefore that, given an inner-analytic function and its set of singularities on the unit circle, as well as the corresponding FC pair of DP Fourier series, we may at once define a whole infinite chain of inner-analytic functions and corresponding DP Fourier series, running by differentiation to one side and by integration to the other, indefinitely in both directions. We will name this an integral-differential chain. We are now ready to give the complete formal definition of the proposed classification of the singularities of inner analytic functions $w(z)$ on the unit circle. Let $z_{1}$ be a point on the unit circle. We start with the very basic classification which was already mentioned.

• A singularity of $w(z)$ at $z_{1}$ is a soft singularity if the limit of $w(z)$ from within the unit disk to that point exists and is finite.

• A singularity of $w(z)$ at $z_{1}$ is a hard singularity if the limit of $w(z)$ from within the unit disk to that point does not exist, or is infinite.

Next we establish a gradation of the concepts of hardness and softness of the singularities of $w(z)$. To each singular point $z_{1}$ we attach an integer giving either its degree of hardness or its degree of softness. In order to do this the following definitions are adopted.

• A single logarithmic integration of $w(z)$, that does not change the hard/soft character of a singularity, increases the degree of softness of that singularity by one, if it is soft, or decreases the degree of hardness of that singularity by one, if it is hard.

• A single logarithmic differentiation of $w(z)$, that does not change the hard/soft character of a singularity, increases the degree of hardness of that singularity by one, if it is hard, or decreases the degree of softness of that singularity by one, if it is soft.

• Given a soft singularity at $z_{1}$, if $w(z)$ can be logarithmically differentiated indefinitely without that singularity ever becoming hard, then we say that it is an infinitely soft singularity. This means that its degree of softness is infinite.

• Given a hard singularity at $z_{1}$, if $w(z)$ can be logarithmically integrated indefinitely without that singularity ever becoming soft, then we say that it is an infinitely hard singularity. This means that its degree of hardness is infinite.

• A singularity of $w(z)$ at $z_{1}$ is a borderline soft one if it is a soft singularity and a single logarithmic differentiation of $w(z)$ results in a hard singularity at that point. A borderline soft singularity has degree of softness zero.

• A singularity of $w(z)$ at $z_{1}$ is a borderline hard one if it is a hard singularity and a single logarithmic integration of $w(z)$ results in a soft singularity at that point. A borderline hard singularity has degree of hardness zero.

Finally, the following rules are adopted regarding the superposition of several singularities as the same point, brought about by the addition of functions.

• If there is more than one hard singularity of $w(z)$ superposed at $z_{1}$, then the result is a hard singularity and its degree of hardness is that of the component singularity with the largest degree of hardness.

• If there is more than one soft singularity of $w(z)$ superposed at $z_{1}$, then the result is a soft singularity and its degree of softness is that of the component singularity with the smallest degree of softness.

• The superposition of hard singularities and soft singularities of $w(z)$ at $z_{1}$ results in a hard singularity at $z_{1}$, with the degree of hardness of the component singularity with the largest degree of hardness.

It is not difficult to see that this classification spans all existing possibilities in so far as the possible types of singularity go. First, given a point of singularity, either the limit of the function to that point from within the open unit disk exists or it does not. There is no third alternative, and therefore every singularity is either soft or hard. Second, given a soft singularity, either it becomes hard after a certain finite number of logarithmic differentiations of $w(z)$, or it does not. Similarly, given a hard singularity, either it becomes soft after a certain finite number of logarithmic integrations of $w(z)$, or it does not. In either case there is no third alternative. If the soft or hard character never changes, then we classify the singularity as infinitely soft or infinitely hard, as the case may be. Otherwise, we assign to it a degree of softness or hardness by counting the number $n$ of logarithmic differentiations or logarithmic integrations required to effect its change of character, and assigning to it the number $n-1$ as the degree of softness or hardness, as the case may be.

We now recall that there is a set of hard singularities which is already classified, by means of the concept of the Laurent expansion around an isolated singular point. If a singularity is isolated in two ways, first in the sense that there is an open neighborhood around it that contains no other singularities, and second that it is possible to integrate along a closed curve around it which is closed in the sense that it does not pass to another leaf of a Riemann surface when it goes around the point, then one may write a convergent Laurent expansion for the function around that point. This leads to the concepts of poles of finite orders and of essential singularities. In particular, it implies that any analytic function that has a pole of finite order at the point $z_{1}$ can be written around that point as the sum of a function which is analytic at that point and a finite linear combination of the singularities

$$\frac{1}{(z-z_{1})^{n}},$$

for $1\leq n\leq n_{\rm o}$, where $n_{\rm o}$ is the order of the pole. We can use this set of singularities to illustrate our classification. For example, if we have an inner analytic function $w(z)$ with a simple pole $1/(z-z_{1})$ for $z_{1}$ on the unit circle, which is a hard singularity with degree of hardness $1$, then the logarithmic derivative of $w(z)$ has a double pole $1/(z-z_{1})^{2}$ at that point, an even harder singularity, of degree of hardness $2$. Further logarithmic differentiations of $w(z)$ produce progressively harder singularities $1/(z-z_{1})^{n}$, where $n$ is the degree of hardness of the singularity. We see therefore that multiple poles fit easily and comfortably into the classification scheme. We may now proceed to examine this chain of singularities in the other direction, using logarithmic integration in order to do this.

The logarithmic primitive of the function $w(z)$ mentioned above has a logarithmic singularity $\ln(z-z_{1})$ at that point, which is the weakest type of hard singularity in this type of integral-differential chain. Another logarithmic integration produces a soft singularity such as $(z-z_{1})\ln(z-z_{1})$, which displays no divergence to infinity. This establishes therefore that $\ln(z-z_{1})$ is a borderline hard singularity, with degree of hardness $0$. If we now proceed to logarithmically differentiate the resulting function, we get back the hard singularity $\ln(z-z_{1})$. This establishes therefore that $(z-z_{1})\ln(z-z_{1})$ is a borderline soft singularity, with degree of softness $0$. This illustrates the transitions between hard and soft singularities, and also justifies our attribution of degrees of hardness to the multiple poles, as we did above. Further logarithmic integrations produce progressively softer singularities such as $(z-z_{1})^{2}\ln(z-z_{1})$, and so on, were we consider only the hardest or least soft singularity resulting from each operation and ignore regular terms, leading to the general expression

$$(z-z_{1})^{n+1}\ln(z-z_{1}),$$

where $n$ is the degree of softness. This completes the examination of the singularities of this particular type of integral-differential chain. Note that these soft singularities are isolated in the sense that there is an open neighborhood around each one of them that contains no other singularities, but not in the sense that one can integrate in closed curves around them. This is so because the domains of these functions are in fact Riemann surfaces with infinitely many leaves, and a curve which is closed in the complex plane is not really closed in the domain of the function.

Although this chain of singularities exhausts the possibilities so far as one is limited to integral-differential chains containing isolated hard singularities of single-valued functions, there are many other possible chains of singularities, if one starts with hard singularities having non-trivial Riemann surfaces, for example such as

$$\frac{\ln(z-z_{1})}{(z-z_{1})^{n}},$$

for $n\geq 1$. One might consider also the more general form

$$(z-z_{1})^{n}\ln^{m}(z-z_{1})$$

for the singularities, where $m\geq 0$ and $n$ in any integer, positive or negative. This generates quite a large set of possible types of singularity, both soft and hard.

To complete the picture in our exemplification, a simple and widely known example of an infinitely hard singularity is an essential singularity such as $\exp[1/(z-z_{1})]$. On the other hand, an infinitely soft singularity is not such a familiar object. One interesting example will be discussed in Section A.2 of Appendix A.

It is important to note that almost all convergent DP Fourier series will be related to inner analytic functions either with only soft singularities on the unit circle or with at most borderline hard singularities, which will therefore all have non-trivial Riemann surfaces as their domains. Since in our analysis here we are bound within the unit disk, and will at most consider limits to the unit circle from within that disk, this is not of much concern to us, because in this case we never go around one of these singularities in order to change from one leaf of the Riemann surface to another. The value of the function $w(z)$ within the unit disk is defined by its value a the origin, and this determines the leaf of each Riemann surface which is to be used within the disk. We must always consider that the branching lines of all such branching points at the unit circle extend outward from the unit circle, towards infinity.