Generalization to Arbitrary Analytic Functions

The basic idea of the proof of the generalization of the extended Cauchy-Goursat theorem will be to embed an arbitrarily given integration contour on the $z_{b}$ plane, which must be a simple closed curve but may or may not be a differentiable curve, into a region bounded by a differentiable simple closed curve, which is then mapped to the unit circle on the $z_{a}$ plane by a conformal transformation. The embedding will be such that all the isolated integrable singularities on the original integration contour are mapped onto the unit circle. This will then allow us to use the extended Cauchy-Goursat theorem for inner analytic functions within the unit disk, which was established in Section 3, to prove the generalized version of that theorem. Therefore, in this section we will prove the following theorem.

Theorem 2   : Given an analytic function $w(z)$, and a closed integration contour $C$ within which $w(z)$ is analytic, and on which $w(z)$ is analytic except for a finite number of isolated integrable singularities, it follows that the integral of $w(z)$ over the contour $C$ is zero,

$$\oint_{C}dz\, w(z) = 0.$$ (24)

This is the most complete generalization of the extended Cauchy-Goursat theorem that we will consider here, the only relevant limitation being that the number of isolated integrable singularities be finite.

Proof 2.1   :

Let $w_{b}(z_{b})$ be an analytic function within a closed simple curve $D_{b}$ on the $z_{b}$ plane, and let the function $w_{b}(z_{b})$ also be analytic almost everywhere on $D_{b}$, with the exception of a finite number of isolated integrable singular points. It follows that, due to Lemmas 1-4, the corresponding function $w_{a}(z_{a})$ on the $z_{a}$ plane will have the same number of singularities on it, which will also be isolated integrable singular points. The curve $D_{b}$ may not be differentiable at some points, including at some of the singular points. We will consider the integral of $w_{b}(z_{b})$ over the integration contour $D_{b}$ on the $z_{b}$ plane, which will then, of course, correspond to the integral of $w_{a}(z_{a})$ over a corresponding integration contour $D_{a}$ on the $z_{a}$ plane, under the conformal transformation.

Figure 5: An integration contour $D_{b}$ with three isolated singularities located at the points $z_{1}$, $z_{2}$ and $z_{3}$, where that contour is non-differentiable and concave, showing how to decompose it into three contours on which the singularities are located at points where the new contours are non-differentiable but convex, by the cuts shown (dashed lines).

The proof that follows will depend on the integration contour $D_{b}$ being either differentiable or non-differentiable and convex at all the singular points found on it. However, this is not a true limitation, because an integration contour that has one of more singular points where it is non-differentiable and concave can always be decomposed into two or more integration contours where those same singular points are convex, as is shown in Figure 5 for an example with three such points. As one can see, all the three integration contours into which the original one is decomposed by the cuts shown (dashed lines) are convex at the singular points. When the three are put together to form once again the original contour, the integrals over those cuts, which due to their orientation are traversed once in each direction, cancel out. Therefore, if the theorem is proven for all contours which are convex at the singular points, it follows that it in fact holds for all contours, regardless of whether they are convex or concave at their singular points. We may therefore limit the proof to contours which are convex at all their singular points.

Given an integration contour $D_{b}$ within which $w_{b}(z_{b})$ is analytic, and on which $w_{b}(z_{b})$ is analytic except for a finite number of isolated integrable singularities, at all of which the contour is either differentiable or non-differentiable and convex, we consider now the construction on the $z_{b}$ plane of a new closed differentiable simple curve $C_{b}$ that contains $D_{b}$. At any points on $D_{b}$ where $w_{b}(z_{b})$ is analytic there is a neighborhood of that point within which there are no singularities of $w_{b}(z_{b})$ which are not located directly on $D_{b}$, whose union forms a strip around $D_{b}$. In this case we make $C_{b}$ go through these neighborhoods in a differentiable fashion, outside the interior of $D_{b}$, so that we ensure that no singularities of $w_{b}(z_{b})$ get included on $C_{b}$ or in its interior, other than those on $D_{b}$. This can be done even at non-singular points where the integration contour $D_{b}$ is not differentiable, in which case we make $C_{b}$ just go around the point of non-differentiability of $D_{b}$, in a differentiable fashion, as can be seen illustrated in Figure 6.

Figure 6: The construction a differentiable closed curve $C_{b}$ containing the integration contour $D_{b}$, to be conformally mapped to the unit circle $C_{a}$, showing also the singular points $z_{1}$ and $z_{2}$, as well as a point of non-differentiability of the contour at which $w(z)$ is analytic.

At points of $D_{b}$ where $w_{b}(z_{b})$ has an isolated integrable singularity, since the singularity is isolated, there is also a neighborhood of the point within which there are no other singularities of $w_{b}(z_{b})$, which is part of the aforementioned strip around $D_{b}$. In this case we make $C_{b}$ go through this neighborhood, still keeping to the outer side of $D_{b}$, in such a way that the curve runs over the singular point in a differentiable way, which is possible because $D_{b}$ is convex at that singular point, as is illustrated by the point $z_{1}$ in Figure 6. The singular point is one which the curve $C_{b}$ will therefore share with $D_{b}$, as is also illustrated by the point $z_{1}$ in Figure 6. At singular points where $D_{b}$ is differentiable we simply make $C_{b}$ tangent to $D_{b}$ at that point, as is illustrated by the point $z_{2}$ in Figure 6.

The result of this process, taken all around $D_{b}$ and including all the isolated integrable singularities found on it, is a differentiable simple curve $C_{b}$ which contains $D_{b}$ and the singularities on it, but that contains no other singularities of $w_{b}(z_{b})$, and which shares with $D_{b}$ all the points where the relevant isolated integrable singularities of $w_{b}(z_{b})$ are located. Since by construction $C_{b}$ is a differentiable closed simple curve, by the Riemann mapping theorem there exists a conformal transformation $\gamma(z_{a})$ that maps it from the unit circle. Therefore, the inverse conformal transformation $\gamma^{(-1)}(z_{b})$ will map all the isolated integrable singular points on $D_{b}$ to the unit circle $C_{a}$.

Since the interior of $C_{b}$ in mapped by the inverse transformation $\gamma^{(-1)}(z_{b})$ onto the open unit disk, it follows that the integration contour $D_{b}$, which is contained in $C_{b}$, is mapped by $\gamma^{(-1)}(z_{b})$ onto a closed simple integration contour $D_{a}$ contained in the unit disk in the $z_{a}$ plane, which will not be differentiable if $D_{b}$ is not, but which is contained within the closed unit disk, and that touches the unit circle only at each one of the isolated integrable singular points of $w_{a}(z_{a})$ on $D_{a}$ that correspond to the singularities of $w_{b}(z_{b})$ on $D_{b}$.

Observe that, since the curve $C_{b}$ does not contain any singularities of $w_{b}(z_{b})$ in its strict interior, the interior of the curve $C_{a}$, which is the open unit disk, does not contain any singularities of $w_{a}(z_{a})$. Therefore, according to the definition given in [#!CAoRFI!#], $w_{a}(z_{a})$ is an inner analytic function. If we now consider the integral of $w_{b}(z_{b})$ over $D_{b}$, it is a very simple thing to change the integration variable from $z_{b}$ to $z_{a}$,

$$\oint_{D_{b}}dz_{b}\, w_{b}(z_{b}) = \oint_{D_{a}}dz_{a}\, \left( \frac{dz_{b}}{dz_{a}} \right) w_{a}(z_{a})$$

$$= \oint_{D_{a}}dz_{a}\, \left[ \frac{d\gamma(z_{a})}{dz_{a}} \right] w_{a}(z_{a}),$$ (25)

where we used the relations shown in Equations (14) and (16). Because $\gamma(z_{a})$ is an analytic function on the whole closed unit disk, the derivative in brackets is also an analytic function on the whole closed unit disk, and in addition to this the function $w_{a}(z_{a})$ is analytic within the integration contour $D_{a}$ and also on $D_{a}$ except for a finite set of isolated singularities located on the unit circle. By the results of Lemmas 1-4, these isolated singularity are all integrable ones. Therefore, since the product of two analytic functions is also an analytic function, the integrand is analytic within the integration contour $D_{a}$, and also on it except for a finite set of isolated integrable singularities on the unit circle, and hence is an inner analytic function. Therefore, by Theorem 1, that is, the extended Cauchy-Goursat theorem for inner analytic functions on the unit disk, the integral is zero, and hence it follows that

$$\oint_{D_{b}}dz_{b}\, w_{b}(z_{b}) = 0.$$ (26)

In other words, due to the fact that the integral of $w_{a}(z_{a})$ on $D_{a}$ is zero, which is guaranteed by the extended Cauchy-Goursat theorem for inner analytic functions, we may conclude that the integral of $w_{b}(z_{b})$ on $D_{b}$ is also zero. This implies that the extended Cauchy-Goursat theorem is valid for $w_{b}(z_{b})$, that is, for arbitrary complex analytic functions anywhere on the complex plane. This completes the proof of Theorem 2.

Note that, once we have Theorem 2 established, it is also valid for all inner analytic functions, and therefore automatically includes the contents of Theorem 1, which we may therefore regard as just an intermediate step in the proof.