Refinement of Two Previous Results

The discussions of two of the results related to the complex-analytic structure introduced in [#!CAoRFI!#], whose proofs were presented in that paper, turn out to be somewhat incomplete for our needs here, so that the proofs presented there must be somewhat refined. These are the discussions regarding the fact that soft singularities must be integrable ones (Property $5.1$ of Definition $5$), and the fact that borderline hard singularities must be integrable ones (Property $8.1$ of Definition $8$). Let us discuss and refine each one in turn.

With regard to the fact that a soft singularity of an inner analytic function $w(z)$ at a point $z_{1}$ on the unit circle must be an integrable one, the discussion given in [#!CAoRFI!#] takes us to the point where it is shown that the integral of $w(z)$ exists on all simple curves contained within the unit disk that connect to the point $z_{1}$. However, we neglected to point out that certain integrals involving $w(z)$ have all the same value, which is equivalent to the fact that the angular primitive $w^{-1\mbox{\Large$\cdot$}\!}(z)$ is well-defined at $z_{1}$. Let us repeat the argument here. Therefore, we now review the following important property of soft singularities, first established in [#!CAoRFI!#].

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

This is so because the angular integration of $w(z)$ produces its angular primitive, an inner analytic function $w^{-1\mbox{\Large$\cdot$}\!}(z)$ which also 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 involving $w(z)$ along a curve from the origin to $z_{1}$, as shown in Equation (2), that integral must therefore exist and result in a finite complex number, for all curves within the open unit disk that go from $0$ to $z_{1}$. Since the other factor involved in the integrand of that integral is a regular function which is different from zero in a neighborhood around $z_{1}$, say an open disk of radius $\varepsilon$ as shown in Figure 1, this implies that $w(z)$ must be integrable around $z_{1}$. Therefore, the singularity of $w(z)$ at $z_{1}$ must be an integrable one.

In addition to this, since $w^{-1\mbox{\Large$\cdot$}\!}(z)$ has a definite finite complex value at $z_{1}$, we may also conclude that the value of the integral giving it does not depend on the integration contour from $0$ to $z_{1}$, that is, on the direction along which that curve connects to $z_{1}$. Then the Cauchy-Goursat theorem in its usual form, which implies that integrals from the origin to any point $z_{0}$ within the open unit disk are independent of the integration contours connecting those two points and contained within the open unit disk, allows us to generalize the result, in a straightforward way, from curves starting at the point $0$ to curves starting at any internal point $z_{0}$ within the open unit disk, by simply connecting the origin and $z_{0}$ by means of any simple curve, for example the straight segment shown in Figure 1. We therefore conclude that, given an integration contour $C$ contained within the unit disk and going from $z_{0}$ to $z_{1}$, the integral

$$\int_{C}dz\, \frac{w(z)-w(0)}{z}$$ (4)

does not depend on the contour. We thus establish this property in a more complete way.

Figure 1: The unit circle of the complex plane, the singular point $z_{1}$, the origin $0$, the internal point $z_{0}$, the neighborhood of radius $\varepsilon$, the integration contour $C$ and the straight segment connecting $z_{0}$ to the origin.

With regard to the fact that a borderline hard singularity of an inner analytic function $w(z)$ at a point $z_{1}$ on the unit circle must be an integrable one, we again neglected to point out in [#!CAoRFI!#] that the same set of integrals from $z_{0}$ to $z_{1}$ discussed in the previous case, along any simple curve contained within the unit disk and connecting those two points, are all equal, which once more is equivalent to the fact that the angular primitive $w^{-1\mbox{\Large$\cdot$}\!}(z)$ is well-defined at $z_{1}$. Let us repeat the argument here. Therefore, we now review the following important property of borderline hard singularities, first established in [#!CAoRFI!#].

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

This is so because the angular integration of $w(z)$ produces its angular primitive, an inner analytic function $w^{-1\mbox{\Large$\cdot$}\!}(z)$ which has at $z_{1}$ a soft singularity, given that the singularity of $w(z)$ at that point is a borderline hard one, and therefore $w^{-1\mbox{\Large$\cdot$}\!}(z)$ is well defined at the point $z_{1}$. Since the value of $w^{-1\mbox{\Large$\cdot$}\!}(z)$ at $z_{1}$ is given by an integral involving $w(z)$ along a curve from the origin to $z_{1}$, as shown in Equation (2), that integral must therefore exist and result in a finite complex number, for all curves within the unit disk that go from $0$ to $z_{1}$. Since the other factor involved in the integrand of that integral is a regular function which is different from zero in the neighborhood around $z_{1}$, this implies that $w(z)$ must be integrable around $z_{1}$. Therefore, the singularity of $w(z)$ at $z_{1}$ must be an integrable one.

In addition to this, since $w^{-1\mbox{\Large$\cdot$}\!}(z)$ has a definite finite complex value at $z_{1}$, we may also conclude that the value of the integral giving it does not depend on the integration contour from the origin to $z_{1}$, that is, on the direction along which that curve connects to $z_{1}$. Just as was noted before in the discussion of the previous case, at this point the Cauchy-Goursat theorem allows us to generalize the result, in a straightforward way, from curves starting at the point $0$ to curves starting at any internal point $z_{0}$ on the open unit disk. We therefore conclude that, given an integration contour $C$ contained within the unit disk and going from $z_{0}$ to $z_{1}$, the integral

$$\int_{C}dz\, \frac{w(z)-w(0)}{z}$$ (5)

does not depend on the contour. We thus establish this property in a more complete way.

We may express these two results, about the singularities being integrable, in a more concise way, by simply stating that what we mean by the integrability of an inner analytic function $w(z)$ around a singular point $z_{1}$ on the unit circle is that the integrals shown in Equation (2) on curves contained within the open unit disk and going from any internal point $z_{0}$ to that singular point both exist and are independent of the curves. This is, of course, equivalent to the statement that the angular primitive of $w(z)$ exists and is well-defined at $z_{1}$.

Once again we recall that, as was noted in [#!CAoRFI!#], whenever the singularities on the unit circle are branch points, it is understood that the corresponding branch cuts are to be extended outward from the unit circle, so that no branch cuts cross the unit disk. In this way there can be no crossings between branch cuts and integration contours within the open unit disk. This simplifies the arguments, since such crossings would force us to consider the fact that the integration contours might be changing from one leaf of a Riemann surface to another. However, the fact that this is not an essential hypothesis is apparent when one considers that for closed integration contours these crossings would necessarily happen in pairs, each pair representing a change of leafs followed by a change back to the original leaf, given that all branch cuts within the open unit disk must cross it completely, since there are no singularities of $w(z)$ within the open unit disk.