Synopsis:

The Complex-Analytic Structure

An inner analytic function $w(z)$ is simply a complex function which is analytic within the open unit disk. An inner analytic function that has the additional property that $w(0)=0$ is a proper inner analytic function. The angular derivative of an inner analytic function is defined by

$$w^{\mbox{\Large$\cdot$}\!}(z) = \mbox{\boldmath$\imath$} z\, \frac{dw(z)}{dz}.$$ (1)

By construction we have that $w^{\mbox{\Large$\cdot$}\!}(0)=0$, for all $w(z)$. The angular primitive of an inner analytic function is defined by

$$w^{-1\mbox{\Large$\cdot$}\!}(z) = -\mbox{\boldmath$\imath$} \int_{0}^{z}dz'\, \frac{w(z')-w(0)}{z'}.$$ (2)

By construction we have that $w^{-1\mbox{\Large$\cdot$}\!}(0)=0$, for all $w(z)$. In terms of a system of polar coordinates $(\rho,\theta)$ on the complex plane, these two analytic operations are equivalent to differentiation and integration with respect to $\theta$, taken at constant $\rho$. These two operations stay within the space of inner analytic functions, they also stay within the space of proper inner analytic functions, and they are the inverses of one another. Using these operations, and starting from any proper inner analytic function $w^{0\mbox{\Large$\cdot$}\!}(z)$, one constructs an infinite integral-differential chain of proper inner analytic functions,

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

Two different such integral-differential chains cannot ever intersect each other. There is a single integral-differential chain of proper inner analytic functions which is a constant chain, namely the null chain, in which all members are the null function $w(z)\equiv 0$.

A general scheme for the classification of all possible singularities of inner analytic functions is established. A singularity of an inner analytic function $w(z)$ at a point $z_{1}$ on the unit circle is a soft singularity if the limit of $w(z)$ to that point exists and is finite. Otherwise, it is a hard singularity. Angular integration takes soft singularities to other soft singularities, and angular differentiation takes hard singularities to other hard singularities.

Gradations of softness and hardness are then established. A hard singularity that becomes a soft one by means of a single angular integration is a borderline hard singularity, with degree of hardness zero. The degree of softness of a soft singularity is the number of angular differentiations that result in a borderline hard singularity, and the degree of hardness of a hard singularity is the number of angular integrations that result in a borderline hard singularity. Singularities which are either soft or borderline hard are integrable ones. Hard singularities which are not borderline hard are non-integrable ones.

Given an integrable real function $f(\theta)$ on the unit circle, one can construct from it a unique corresponding inner analytic function $w(z)$. The real function $f(\theta)$ is recovered by means of the $\rho\to 1_{(-)}$ limit of the real part of this inner analytic function. Singularities of real functions can be classified in a way which is analogous to the corresponding complex classification. Integrable real functions are typically associated with inner analytic functions that have singularities which are either soft or at most borderline hard. A more detailed review of real functions will be given in Section 2. This ends our synopsis.

Some of the material contained in this paper can be seen as a development, reorganization and extension of some of the material found, sometimes still in rather rudimentary form, in the papers [#!FTotCPI!#,#!FTotCPII!#,#!FTotCPIII!#,#!FTotCPIV!#,#!FTotCPV!#].