Definitions and Properties

Here we will introduce the definitions and basic properties of some objects and structures which are not usually discussed in complex analysis [#!CVchurchill!#], and which we will use in the subsequent sections. Consider then the unit circle centered at the origin of the complex plane. Its interior is the open unit disk we will often refer to along the paper. Any reference to the unit disk or to the unit circle should always be understood to refer to those centered at the origin.

Definition 1   : Inner Analytic Functions

A complex function $w(z)$ which is analytic in the open unit disk will be named an inner analytic function. We will consider the set of all such functions. We will also consider the subset of such functions that have the additional property that $w(0)=0$, which we will name proper inner analytic functions.

Note, in passing, that the set of all inner analytic functions forms a vector space over the field of complex numbers, and so does the subset of all proper inner analytic functions.

The focus of this study will be the set of real objects which are obtained from the real parts of these inner analytic functions when we take the limit from the open unit disk to its boundary, that is, to the unit circle. Specifically, if we describe the complex plane with polar coordinates $(\rho,\theta)$, then an arbitrary inner analytic function can be written as

$$w(z) = u(\rho,\theta)+\mbox{\boldmath$\imath$}v(\rho,\theta),$$ (1)

where $\mbox{\boldmath$\imath$}$ is the imaginary unit, and we consider the set of real objects $f(\theta)$ obtained from the set of all inner analytic functions as the limits of their real parts, from the open unit disk to the unit circle,

$$f(\theta) = \lim_{\rho\to 1_{(-)}}u(\rho,\theta),$$ (2)

when and where such limits exist, or at least can be defined in a consistent way.

Note that an inner analytic function may have any number of singularities on the unit circle, as well as in the region outside the unit circle. The concept of a singularity is the usual one in complex analysis, namely that a singular point is simply a point where the function fails to be analytic. The singularities on the unit circle will play a particularly important role in the complex-analytic structure to be presented in this paper. If any of these singularities turn out to be branch points, then we assume that the corresponding branch cuts extend outward from the unit circle, either out to infinity or connecting to some other singularity that may exist outside the open unit disk.

Note also that the imaginary parts of the inner analytic functions do not generate an independent set of real objects, since the imaginary part $v(\rho,\theta)$ of the inner analytic function $w(z)$ is also the real part of the inner analytic function $\bar{w}(z)$ given by

$$\bar{w}(z) = -\mbox{\boldmath$\imath$}w(z).$$ (3)

We thus see, however, that the inner analytic functions do organize the real functions in matched pairs, those originating from the real and imaginary parts of each inner analytic function. The two real functions forming such a pair may be described as mutually Fourier conjugate functions. Finally, we will assume that, at all singular points where the functions $w(z)$ can still be defined by continuity, they have been so defined.

In addition to establishing this correspondence between complex functions on the unit disk and real function of the unit circle, we will find it necessary to define analytic operations on the complex functions that correspond to the ordinary operations of differentiation and integration on the real functions. As will be shown in what follows, the next two definitions accomplish this.

Definition 2   : Angular Differentiation

Given an arbitrary inner analytic function $w(z)$, its angular derivative is defined by

$$w^{\mbox{\Large$\cdot$}\!}(z) = \mbox{\boldmath$\imath$} z\, \frac{dw(z)}{dz}.$$ (4)

The angular derivative of $w(z)$ will be denoted by the shifted dot, as shown. The second angular derivative will be denoted by $w^{2\mbox{\Large$\cdot$}\!}(z)$, and so on.

Note that this definition has been tailored in order for the following property to hold.

Property 2.1   : In terms of the variables $(\rho,\theta)$, angular differentiation is equivalent to partial differentiation with respect to $\theta$, taken at constant $\rho$.

Writing $z=\rho\exp(\mbox{\boldmath$\imath$}\theta)$, and considering the partial derivative of $z$ with respect to $\theta$, we have

$$w^{\mbox{\Large$\cdot$}\!}(z) = \mbox{\boldmath$\imath$}\rho\,{\rm e}^{\mbox{\boldmath\scriptsize... ...ox{\boldmath\scriptsize$\imath$}\theta}}\, \frac{\partial w(z)}{\partial\theta}$$

$$= \frac{\partial u(\rho,\theta)}{\partial\theta} + \mbox{\boldmath$\imath$}\, \frac{\partial v(\rho,\theta)}{\partial\theta},$$ (5)

which establishes this property.

Note that by construction we always have that $w^{\mbox{\Large$\cdot$}\!}(0)=0$, so that we may say that the operation of angular differentiation projects the space of inner analytic functions onto the space of proper inner analytic functions. We may now prove an important property of the angular derivative.

Property 2.2   : The angular derivative of an inner analytic function is also an inner analytic function.

Let us recall that the derivative of an analytic function always exists and is also analytic, in the same domain of analyticity of the original function. Since the constant function $w(z)\equiv\mbox{\boldmath$\imath$}$ and the identity function $w(z)\equiv z$ are analytic in the whole complex plane, and since the product of analytic functions is also an analytic function, in their common domain of analyticity, it follows at once that the angular derivative of an inner analytic function is an inner analytic function as well, which establishes this property.

In other words, the operation of angular differentiation stays within the space of inner analytic functions. Note that, since $w^{\mbox{\Large$\cdot$}\!}(0)=0$, angular differentiation always results in proper inner analytic functions, and therefore that this operation also stays within the space of proper inner analytic functions.

Definition 3   : Angular Integration

Given an arbitrary inner analytic function $w(z)$, its angular primitive is defined by

$$w^{-1\mbox{\Large$\cdot$}\!}(z) = -\mbox{\boldmath$\imath$} \int_{0}^{z}dz'\, \frac{w(z')-w(0)}{z'},$$ (6)

where the integral is taken along any simple curve from $0$ to $z$ contained within the open unit disk. Since the integrand is analytic inside the open unit disk, including at the origin, as we will see shortly while proving Property 3.2, due to the Cauchy-Goursat theorem the integral does not depend on the curve along which it is taken. The angular primitive will also be denoted by a shifted dot, this time preceded by a negative integer, as indicated above.

Let us prove that the apparent singularity of the integrand at $z=0$ is in fact a removable singularity, so that the integrand can be defined at the origin by continuity, thus producing a function which is continuous and well defined there. If we simply take the $z\to 0$ limit of the integrand we get

$$\lim_{z\to 0} \frac{w(z)-w(0)}{z} = \frac{dw}{dz}(0),$$ (7)

since this limit is the very definition of the derivative of $w(z)$ at $z=0$. Since $w(z)$ is an inner analytic function, and is thus analytic in the open unit disk, it is differentiable at the origin, so that this limit exists and is finite. Therefore the integrand can be defined at the origin to have this particular value, so that it is continuous there. We assume that the integrand is so defined at $z=0$, as part of the definition of the angular primitive.

Note that this definition has been tailored in order for the following property to hold.

Property 3.1   : In terms of the variables $(\rho,\theta)$, angular integration can be understood as integration with respect to $\theta$, taken at constant $\rho$, up to an integration constant.

Given any point $z$ in the open unit disk, and considering that we are free to choose the path of integration from $0$ to $z$, we now choose to go first from the origin along the positive real axis, until we reach the radius $\rho$, and then to go along an arc of circle of radius $\rho$, until we reach the angle $\theta$, thus separating the integral in two. In the first integral the variations of $z$ are given by $dz=d\rho$, and in the second one they are given by $dz=\mbox{\boldmath$\imath$}\rho\exp(\mbox{\boldmath$\imath$}\theta)d\theta$. Note that as the integrand in Equation (6) we have the proper inner analytic function given by

$$w_{p}(z) = w(z)-w(0)$$

$$= u_{p}(\rho,\theta) + \mbox{\boldmath$\imath$} v_{p}(\rho,\theta),$$ (8)

where $w_{p}(0)=0$. The integral in Equation (6) can therefore be written as

$$w^{-1\mbox{\Large$\cdot$}\!}(z) = -\mbox{\boldmath$\imath$} \int_{0}^{\rho}d\rho'\, \frac{w_{p}(\rh... ..._{p}(\rho,\theta')}{\rho\,{\rm e}^{\mbox{\boldmath\scriptsize$\imath$}\theta'}}$$

$$= C(\rho) + \int_{0}^{\theta}d\theta'\, u_{p}(\rho,\theta') + \mbox{\boldmath$\imath$} \int_{0}^{\theta}d\theta'\, v_{p}(\rho,\theta'),$$ (9)

where, in relation to the variable $\theta$, the integral on $\rho'$ becomes the complex function $C(\rho)$, which depends only on $\rho$ and not on $\theta$, while the integral on $\theta'$ determines primitives with respect to $\theta$ of the real and imaginary parts of $w_{p}(z)$, which thus establishes this property.

Note that by construction we always have that $w^{-1\mbox{\Large$\cdot$}\!}(0)=0$, so that we may say that the operation of angular integration projects the space of inner analytic functions onto the space of proper inner analytic functions. We may now establish an important property of the angular primitive.

Property 3.2   : The angular primitive of an inner analytic function is also an inner analytic function.

In order to prove that $w^{-1\mbox{\Large$\cdot$}\!}(z)$ is an inner analytic function, we use the power-series representation of the inner analytic function $w(z)$. Since this function is analytic within the open unit disk, its Taylor series around $z=0$, which is given by

$$w(z) = w(0) + \sum_{k=1}^{\infty} c_{k}z^{k},$$ (10)

where $c_{k}=w^{k\prime}(0)/k!$ are the Taylor coefficients of $w(z)$ with respect to the origin and where $w^{k\prime}(z)$ is the $k^{\rm th}$ derivative of $w(z)$, converges within that disk. We therefore have for the integrand in Equation (6) the power-series representation

$$\frac{w(z)-w(0)}{z} = \sum_{k=1}^{\infty} c_{k}z^{k-1}$$

$$= \sum_{k=0}^{\infty} c_{k+1}z^{k}.$$ (11)

Since this series has the same set of coefficients as the convergent series of $w(z)$, it is equally convergent, as is implied for example by the ratio test. Note that this shows, in particular, that the integrand is analytic at $z=0$. Being a convergent power series, this series can be integrated term by term, resulting in an equally convergent power series, so that we have for the angular primitive

$$w^{-1\mbox{\Large$\cdot$}\!}(z) = -\mbox{\boldmath$\imath$} \int_{0}^{z}dz'\, \sum_{k=0}^{\infty} c_{k+1}(z')^{k}$$

$$= -\mbox{\boldmath$\imath$} \sum_{k=0}^{\infty} \frac{c_{k+1}}{k+1}\, z^{k+1}$$

$$= -\mbox{\boldmath$\imath$} \sum_{k=1}^{\infty} \frac{c_{k}}{k}\, z^{k}.$$ (12)

Due to the factors of $1/k$, when $k\to\infty$ the coefficients of this series go to zero faster than those of the convergent Taylor series of $w(z)$, and thus it is also convergent, in the same domain of convergence of the Taylor series of $w(z)$. This confirms that this series is convergent within the open unit disk. Being a convergent power series, it converges to an analytic function, thus proving that $w^{-1\mbox{\Large$\cdot$}\!}(z)$ is analytic within the open unit disk. We may conclude therefore that the angular primitive of an inner analytic function is an inner analytic function as well, which establishes this property.

In other words, the operation of angular integration stays within the space of inner analytic functions. Note that, since $w^{-1\mbox{\Large$\cdot$}\!}(0)=0$, angular integration always results in proper inner analytic functions, and therefore that this operation also stays within the space of proper inner analytic functions.

Let us now prove that the operations of angular differentiation and of angular integration are inverse operations to one another. Strictly speaking, this is true within the subset of inner analytic functions that have the additional property that $w(0)=0$, that is, for proper inner analytic functions. Since any inner analytic function can be obtained from a proper inner analytic function by the mere addition of a constant, this is a very weak limitation. Let us consider then the space of proper inner analytic functions.

Property 3.3   : The angular primitive of the angular derivative of a proper inner analytic function is that same proper inner analytic function.

We simply compose the two operations in the required order, and calculate in a straightforward manner, merely using the fundamental theorem of the calculus, to get

$$-\mbox{\boldmath$\imath$} \int_{0}^{z}dz'\, \frac{1}{z'}\, \left[... ...'\, \frac{dw}{dz'}(z') - \mbox{\boldmath$\imath$} 0\, \frac{dw}{dz'}(0) \right] = \int_{0}^{z}dz'\, \frac{dw}{dz'}(z')$$

$$= w(z)-w(0),$$ (13)

which is the original inner analytic function $w(z)$ so long as $w(0)=0$, that is, for proper inner analytic functions, thus establishing this property.

Property 3.4   : The angular derivative of the angular primitive of a proper inner analytic function is that same proper inner analytic function.

We simply compose the two operations in the required order, and calculate in a straightforward manner, using this time the power-series representation of the inner analytic function $w(z)$. First integrating term by term and then differentiating term by term, both of which are allowed operations for convergent power series, we get

$$\mbox{\boldmath$\imath$} z\, \frac{d}{dz} (-\mbox{\boldmath$\imath$}) \int_{0}^{z}dz'\, \frac{w(z')-w(0)}{z'} = z\, \frac{d}{dz} \int_{0}^{z}dz'\, \sum_{k=1}^{\infty} c_{k}(z')^{k-1}$$

$$= z\, \frac{d}{dz} \sum_{k=1}^{\infty} \frac{c_{k}}{k}\,z^{k}$$

$$= \sum_{k=1}^{\infty} c_{k}z^{k}$$

$$= w(z)-w(0),$$ (14)

which is the original inner analytic function $w(z)$ so long as $w(0)=0$, that is, for proper inner analytic functions, thus establishing this property.

With the use of the operations of angular differentiation and angular integration the space of proper inner analytic functions can now be organized as a collection of infinite discrete chains of functions, so that within each chain the functions are related to each other by either angular integrations or angular differentiations. This leads to the definition that follows.

Definition 4   : Integral-Differential Chains

Starting from an arbitrary proper inner analytic function $w(z)$, also denoted as $w^{0\mbox{\Large$\cdot$}\!}(z)$, one proceeds in the differentiation direction to the functions $w^{1\mbox{\Large$\cdot$}\!}(z)$, $w^{2\mbox{\Large$\cdot$}\!}(z)$, $w^{3\mbox{\Large$\cdot$}\!}(z)$, etc, and in the integration direction to the functions $w^{-1\mbox{\Large$\cdot$}\!}(z)$, $w^{-2\mbox{\Large$\cdot$}\!}(z)$, $w^{-3\mbox{\Large$\cdot$}\!}(z)$, etc. One thus produces an infinite chain of proper inner analytic functions such as

$$\left\{ \ldots, w^{-3\mbox{\Large$\cdot$}\!}(z), w^{-2\m... ...}\!}(z), w^{3\mbox{\Large$\cdot$}\!}(z), \ldots\; \right\},$$ (15)

in which angular differentiation takes one to the right and angular integration takes one to the left. We name such a structure an integral-differential chain of proper inner analytic functions. We may refer to the proper inner analytic functions forming the chain as links in that chain.

Note that all the functions in such a chain have exactly the same set of singular points on the unit circle, although the character of these singularities will change from function to function along the chain. Note also that each such integral-differential chain induces, by means of the real parts of their inner analytic functions, a corresponding chain of real objects over the unit circle, when and where the limits from the open unit disk to the unit circle exist, or can be consistently defined. Finally note that, given a singularity at a certain point on the unit circle, the integral-differential chain also induces a corresponding chain of singularities at that point. Let us now prove an important property of these chains, namely that they do not intersect each other.

Property 4.1   : Two different integral-differential chains of proper inner analytic functions cannot have a member-function in common.

In order to prove this, we start by proving that, if two proper inner analytic functions have the same angular derivative, then they must be equal. If we have two such proper inner analytic functions $w_{1}(z)$ and $w_{2}(z)$, the statement that they have the same angular derivative is expressed as

$$\mbox{\boldmath$\imath$} z\, \frac{d}{dz}w_{2}(z) - \mbox{\boldmath$\imath$} z\, \frac{d}{dz}w_{1}(z) = 0 \;\;\;\Rightarrow$$

$$\frac{d}{dz}[w_{2}(z)-w_{1}(z)] = 0 \;\;\;\Rightarrow$$

$$w_{2}(z)-w_{1}(z) = C,$$ (16)

where $C$ is some complex constant, for all $z$ within the open unit disk, including the case $z=0$, as one can see if one takes the limit $z\to 0$ of the last equation above. However, since at $z=0$ we have that $w_{1}(0)=0$ and $w_{2}(0)=0$, it then follows that $C=0$, so that we may conclude that

$$w_{2}(z) \equiv w_{1}(z),$$ (17)

thus proving the point. A similar result is valid for two proper inner analytic functions that have the same angular primitive. Since we have already shown that angular integration and angular differentiation are inverse operations to each other, we can prove this by simply noting the trivial fact that the operation of angular differentiation cannot produce two different results for the same function. Therefore, there cannot exist two different proper inner analytic functions whose angular primitives are one and the same function.

We may now conclude that two different integral-differential chains of proper inner analytic functions can never have a member-function in common, because this would mean that two different proper inner analytic functions would have either the same angular derivative or the same angular primitive, neither of which is possible. It follows that each proper inner analytic function appears in one and only one of these integral-differential chains, which establishes this property.

Note, for future use, that there is a single integral-differential chain of proper inner analytic functions which is a constant chain, in the sense that all member-functions of the chain are equal, namely the null chain, in which all members are the null function $w(z)\equiv 0$. It is easy to verify that the differential equation $w^{\mbox{\Large$\cdot$}\!}(z)=w(z)$ has no other inner analytic function as a solution. Note also that one may consider all the non-proper inner analytic functions $w(z)$ which are related to a given proper inner analytic function $w_{p}(z)$ to also belong to the same link of the corresponding integral-differential chain. Since all such functions have the form

$$w(z) = C + w_{p}(z),$$ (18)

where $C$ is a complex constant, this has the effect of associating to each link of the integral-differential chain of $w_{p}(z)$ a complex plane of constants $C$ in which each point corresponds to a function $w(z)$. In particular, all constant functions are associated to the null function, and therefore to a complex plane of constants at each link of the null chain.

We will now establish a general scheme for the classification of all possible singularities of inner analytic functions. This can be done for analytic functions in general, but we will do it here in a way that is particularly suited for our inner analytic functions.

Definition 5   : Classification of Singularities: Soft and Hard

Let $z_{1}$ be a point on the unit circle. A singularity of an inner analytic function $w(z)$ at $z_{1}$ is a soft singularity if the limit of $w(z)$ to that point exists and is finite. Otherwise, it is a hard singularity.

This is a complete classification of all possible singularities because, 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. We may now establish the following important property of soft singularities.

Property 5.1   : A soft singularity of an inner analytic function $w(z)$ at a point $z_{1}$ of the unit circle is necessarily an integrable one.

In order to prove this first note that, since the singularity at $z_{1}$ is soft, the function $w(z)$ is defined by continuity there, being therefore continuous at $z_{1}$. Consider now a curve contained within the open unit disk, that connects to $z_{1}$ along some direction, that has a finite length, and which is an otherwise arbitrary curve. We have at once that $w(z)$ is analytic at all points on this curve except $z_{1}$. It follows that $w(z)$ is continuous, and thus that $\vert w(z)\vert$ is continuous, everywhere on this curve, including at $z_{1}$. Hence, the limits of $\vert w(z)\vert$ to all points on this curve exist and are finite positive real numbers.

We now note that this set of finite real numbers must be bounded, because otherwise there would be a hard singularity of $w(z)$ somewhere within the open unit disk, where this function is in fact analytic. We conclude therefore that over the curve the function $w(z)$ is a bounded continuous function on a finite-length domain, which implies that $w(z)$ is integrable in that domain. Therefore, we may state that $w(z)$ is integrable along arbitrary curves reaching the point $z_{1}$ from strictly within the open unit disk², which thus establishes this property.

We will now prove a couple of important further properties of the singularity classification, one for soft singularities and one for hard singularities. For this purpose, let $z_{1}$ be a point on the unit circle. Let us discuss first a property of soft singularities, which is related to angular integration.

Property 5.2   : If an inner analytic function $w(z)$ has a soft singularity at $z_{1}$, then the angular primitive of $w(z)$ also has a soft singularity at that point.

In order to prove this we use the fact that a soft singularity is necessarily an integrable one. We already know that if $w(z)$ has a singularity at $z_{1}$, then so does its angular primitive $w^{-1\mbox{\Large$\cdot$}\!}(z)$. If we now consider the angular primitive of $w(z)$ at $z=z_{1}$, we have

$$w^{-1\mbox{\Large$\cdot$}\!}(z_{1}) = -\mbox{\boldmath$\imath$} \int_{0}^{z_{1}}dz\, \frac{w(z)-w(0)}{z},$$ (19)

where the integral can be taken over any simple curve within the open unit disk. We already know that the integrand is regular at the origin. Since the singularity of $w(z)$ at $z_{1}$ is soft, that singularity is integrable along any simple curve within the open unit disk that goes from $z=0$ to $z=z_{1}$. Therefore, it follows that this integral exists and is finite, and thus that $w^{-1\mbox{\Large$\cdot$}\!}(z_{1})$ exists and is finite. Since the function $w^{-1\mbox{\Large$\cdot$}\!}(z)$ is thus well defined³ at $z_{1}$, as well as analytic around that point, it follows that the singularity of $w^{-1\mbox{\Large$\cdot$}\!}(z)$ at $z_{1}$ is also soft, which thus establishes this property.

Let us discuss now a property of hard singularities, which is related to angular differentiation.

Property 5.3   : If an inner analytic function $w(z)$ has a hard singularity at $z_{1}$, then the angular derivative of $w(z)$ also has a hard singularity at that point.

If $w(z)$ has a hard singularity at $z_{1}$, then it is not well defined there, implying that it is not continuous there, and therefore that it is also not differentiable there. This clearly implies that the angular derivative $w^{\mbox{\Large$\cdot$}\!}(z)$ of $w(z)$, which we already know to also have a singularity at $z_{1}$, is not well defined there as well. This in turn implies that the singularity of $w^{\mbox{\Large$\cdot$}\!}(z)$ at $z_{1}$ must be a hard one.

However, the simplest way to prove this property is to note that it follows from the previous one, that is, from Property 5.2. We can prove it by reductio ad absurdum, using the fact that, as we have already shown, the operations of angular differentiation and angular integration are inverse operations to each other. If we assume that $w(z)$ has a hard singularity at $z_{1}$ and that $w^{\mbox{\Large$\cdot$}\!}(z)$ has a soft singularity at that point, then we have an inner analytic function, namely $w^{\mbox{\Large$\cdot$}\!}(z)$, that has a soft singularity at $z_{1}$, while its angular primitive, namely $w(z)$, has a hard singularity at that point. However, according to Property 5.2 this is impossible, since angular integration always takes a soft singularity to another soft singularity. This establishes, therefore, that this property holds.

The use of the operations of angular differentiation and of angular integration now leads to a refinement of our general classification of singularities. We will use them to assign to each singularity a degree of softness or a degree of hardness. Let $w(z)$ be an inner analytic function and $z_{1}$ a point on the unit circle, and consider the following two definitions.

Definition 6   : Classification of Singularities: Gradation of Soft Singularities

Let us assume that $w(z)$ has a soft singularity at $z_{1}$. If an arbitrarily large number of successive angular differentiations of $w(z)$ always results in a singularity at $z_{1}$ which is still soft, then we say that the singularity of $w(z)$ at $z_{1}$ is an infinitely soft singularity. Otherwise, if $n_{s}$ is the minimum number of angular differentiations that have to be applied to $w(z)$ in order for the singularity at $z_{1}$ to become a hard one, then we define $n_{s}$ as the degree of softness of the original singularity of $w(z)$ at $z_{1}$. Therefore, a degree of softness is an integer $n_{s}\in\{1,2,3,\ldots,\infty\}$.

Definition 7   : Classification of Singularities: Gradation of Hard Singularities

Let us assume that $w(z)$ has a hard singularity at $z_{1}$. If an arbitrarily large number of successive angular integrations of $w(z)$ always results in a singularity at $z_{1}$ which is still hard, then we say that the singularity of $w(z)$ at $z_{1}$ is an infinitely hard singularity. Otherwise, if $n_{h}+1$ is the minimum number of angular integrations that have to be applied to $w(z)$ in order for the singularity at $z_{1}$ to become a soft one, then we define $n_{h}$ as the degree of hardness of the original singularity of $w(z)$ at $z_{1}$. Therefore, a degree of hardness is an integer $n_{h}\in\{0,1,2,3,\ldots,\infty\}$.

In order to see that this establishes a complete classification of all possible singularities, let us examine all the possible outcomes when we apply angular differentiations and angular integrations to inner analytic functions. We already saw that, if we apply a angular integration to a soft singularity, then the result is always another soft singularity. Similarly we saw that, if we apply a angular differentiation to a hard singularity, then the result is always another hard singularity. The two remaining alternatives are the application of a angular integration to a hard singularity, and the application of a angular differentiation to a soft singularity. In these two cases the resulting singularity may be either soft or hard, and the remaining possibilities were dealt with in Definitions 6 and 7. Since this applies to all singularities in all integral-differential chains, it applies to all possible singularities of all inner analytic functions.

In some cases examples of this classification are well known. For instance, a simple example of an infinitely hard singularity is any essential singularity. Examples of infinitely soft singularities are harder to come by, and they are related to integrable real functions which are infinitely differentiable but not analytic. A simple example of a hard singularity with degree of hardness $n\geq 1$ is a pole of order $n$. Examples of soft singularities are the square root, and products of strictly positive powers with the logarithm.

If a singularity at a given singular point $z_{1}$ on the unit circle is either infinitely soft or infinitely hard, then the corresponding integral-differential chain of singularities contains either only soft singularities or only hard singularities. If the singularity is neither infinitely soft nor infinitely hard, then at some point along the corresponding integral-differential chain the character of the singularity changes, and from that point on the soft or hard character remains constant at the new value throughout the rest of the integral-differential chain in that direction. Therefore, in each integral-differential chain that does not consist of either only soft singularities or only hard singularities, there is a single transition between two functions on the chain where the character of the singularity changes.

Let us examine in more detail the important intermediary case in which we assign to the singularity the degree of hardness zero, which we will also describe as that of a borderline hard singularity.

Definition 8   : Classification of Singularities: Borderline Hard Singularities

Given an inner analytic function $w(z)$ and a point $z_{1}$ on the unit circle where it has a hard singularity, if a single angular integration of $w(z)$ results in a function $w^{-1\mbox{\Large$\cdot$}\!}(z)$ which has at $z_{1}$ a soft singularity, then we say that the original function $w(z)$ has at $z_{1}$ a borderline hard singularity, that is, a hard singularity with degree of hardness zero.

We establish now the following important property of borderline hard singularities.

Property 8.1   : A borderline hard singularity of an inner analytic function $w(z)$ at $z_{1}$ must be an integrable one.

This is so because the angular integration of $w(z)$ produces an inner analytic function $w^{-1\mbox{\Large$\cdot$}\!}(z)$ which has at $z_{1}$ a soft singularity, and therefore is well defined at that point. Since the value of $w^{-1\mbox{\Large$\cdot$}\!}(z)$ at $z_{1}$ is given by an integral of $w(z)$ along a curve reaching that point, that integral must therefore exist and result in a finite complex number⁴. Therefore, the singularity of $w(z)$ at $z_{1}$ must be an integrable one. We may thus conclude that all borderline hard singularities are integrable ones, which establishes this property.

The transition between a borderline hard singularity and a soft singularity is therefore the single point of transition of the soft or hard character of the singularities along the corresponding integral-differential chain. Starting from a borderline hard singularity, $n_{s}$ angular integrations produce a soft singularity with degree of softness $n_{s}$, and $n_{h}$ angular differentiations produce a hard singularity with degree or hardness $n_{h}$. Note that a strictly positive degree of softness given by $n$ can be identified with a negative degree of hardness given by $-n$, and vice-versa. A simple example of a borderline hard singularity is a logarithmic singularity.

Let us end this section with one more important property of hard singularities.

Property 8.2   : The borderline hard singularities are the only hard singularities that are integrable.

If a hard singularity of $w(z)$ at $z_{1}$ has a degree of hardness of one or larger, then by angular integration it is mapped to another hard singularity, the hard singularity of $w^{-1\mbox{\Large$\cdot$}\!}(z)$ at $z_{1}$. If the hard singularity of $w(z)$ were integrable, then $w^{-1\mbox{\Large$\cdot$}\!}(z)$ would be well defined at $z_{1}$, and therefore its singularity would be soft rather than hard. Since we know that the singularity of $w^{-1\mbox{\Large$\cdot$}\!}(z)$ is hard, it follows that the singularity of $w(z)$ cannot be integrable. In other words, all hard singularities with strictly positive degrees of hardness are necessarily non-integrable singularities, which establishes this property.

Note that the structure of the integral-differential chains establishes within the space of proper inner analytic functions what may be described as a structure of discrete fibers, in which the whole space is decomposed as a set of non-intersecting discrete linear structures. The same is then true for the corresponding real objects on the unit circle. One can then reconstruct the space of all inner analytic functions by associating to each link of each integral-differential chain a complex plane of constants to be added to the proper inner analytic function of that link, in order to get all the non-proper inner analytic functions associated to it. In terms of the corresponding real functions, this corresponds to the association to each link of a real line of real constants.