Extension of the Cauchy-Goursat Theorem

In either one of the two situations examined in Section 2, in which the inner analytic function $w(z)$ has either a soft singularity or a borderline hard singularity at a point $z_{1}$ on the unit circle, we discovered that, given an integration contour $C$ contained within the unit disk and going from any internal point $z_{0}$ to $z_{1}$, the integral

$$\int_{C}dz\, \frac{w(z)-w(0)}{z}$$ (6)

does not depend on the contour. We now observe that, if $w(z)$ is any inner analytic function, then $w_{p}(z)=w(z)-w(0)$ is a proper inner analytic function, since $w_{p}(0)=0$. Therefore what we have here is the statement that the integral

$$\int_{C}dz\, \frac{w_{p}(z)}{z}$$ (7)

does not depend on the contour, for all integration contours $C$ contained within the unit disk that go from $z_{0}$ to $z_{1}$, and for all proper inner analytic functions within the unit disk that have an integrable singularity at $z_{1}$. Let us now consider the integral

$$\int_{C}dz\, w(z),$$ (8)

for an arbitrary inner analytic function $w(z)$ that has an integrable singularity at $z_{1}$. Since $w(z)$ is necessarily regular at $z=0$, it follows that the function $w_{p}(z)=zw(z)$ is a proper inner analytic function, given that $w_{p}(0)=0$. In addition to this, since the function $z$ is analytic everywhere, it also follows that $w_{p}(z)$ and $w(z)$ have the same singularity structure, and thus we can write

$$\int_{C}dz\, w(z) = \int_{C}dz\, \frac{w_{p}(z)}{z},$$ (9)

where by the statement involving Equation (7) this last integral is independent of the integration contour $C$. Therefore, we have the statement that

$$\int_{C}dz\, w(z)$$ (10)

is independent of the contour, for all integration contours $C$ within the unit disk that go from $z_{0}$ to $z_{1}$, and for all inner analytic functions that have an integrable singularity at $z_{1}$.

Figure 2: The unit circle of the complex plane, the singular point $z_{1}$, the internal point $z_{0}$, and the integration contours involved in the proof of the extended Cauchy-Goursat theorem for inner analytic function within the unit disk.

We have thus determined that in these two cases the integral of an arbitrary inner analytic function $w(z)$ from an arbitrary point $z_{0}$, internal to the open unit disk, to the point $z_{1}$ on the unit circle, where in either case $w(z)$ has an isolated integrable singularity, along an integration contour contained within the closed unit disk and that touches the unit circle only at $z_{1}$, is independent of that integration contour from $z_{0}$ to $z_{1}$. Therefore, given two different such curves, such as the curves $C_{1}$ and $C_{2}$ illustrated in Figure 2, we may immediately conclude that the integral of $w(z)$ over the closed integration contour $C$ formed by the two curves is zero,

$$\oint_{C}dz\, w(z) = 0.$$ (11)

Since $z_{0}$ is an arbitrary internal point, this is valid for all closed simple curves $C$ within the unit disk, that touch the unit circle only at $z_{1}$. Observe that what we have concluded here is, in fact, that the validity of the Cauchy-Goursat theorem, for the case of inner analytic functions, is not disturbed by the presence of an isolated singularity on the integration contour, so long as this singularity is either a soft one or a borderline hard one or, in other words, so long as singularities like this are all integrable ones. It is quite clear, therefore, that this result constitutes an extension of the usual form of the Cauchy-Goursat theorem, one which is valid at least for inner analytic functions within the unit disk.

Figure 3: The unit circle of the complex plane, the integration contour $C$ (solid line) with two singular points $z_{1}$ and $z_{2}$, showing how it can be decomposed into two integration contours $C_{1}$ and $C_{2}$, each one with only one singular point, by a simple cut (dashed line).

This result for a single point of singularity can then be trivially extended, by contour manipulation and the repeated use of the Cauchy-Goursat theorem in its usual form, to integration contours that touch the unit circle on a finite number of points, at all of which $w(z)$ has isolated integrable singularities. Therefore, as a side effect of the arguments presented in Section 2, in this section we will prove the following theorem.

Theorem 1   : Given an inner analytic function $w(z)$, and a closed integration contour $C$ contained within the closed unit disk, that touches the unit circle only at a set of points satisfying one of two conditions, either points where $w(z)$ is analytic, or points where $w(z)$ has isolated integrable singularities, of which there must be a finite number, it follows that the integral of $w(z)$ over the contour $C$ is zero,

$$\oint_{C}dz\, w(z) = 0.$$ (12)

Proof 1.1   :

We start with contours that touch the unit circle at a single integrable singular point $z_{1}$, such as the one shown in Figure 2, and by just using the results of Section 2, as expressed by Equation (11), to simply state that the integral of $w(z)$ over any such contour is zero. Next, given a closed contour $C$ that touches the unit circle at two separate singular points $z_{1}$ and $z_{2}$, such as the one shown in Figure 3, it can always be separated into two closed contours $C_{1}$ and $C_{2}$, each one of which touches the unit circle at only one singular point, as in the example shown in Figure 3, by a simple cut (dashed line). When the two separate closed contours $C_{1}$ and $C_{2}$ are joined together to form the complete contour $C$, due to the orientation of the contours the integrals over the cut, which is traversed twice, once in each direction, cancel out. This can also be done for a contour that touches the unit circle at any finite number of separate singular points.

Figure 4: The unit circle of the complex plane, the singular points $z_{1}$, $z_{2}$ and $z_{3}$, and the corresponding integration contours $C_{1}$, $C_{2}$ and $C_{3}$ (dashed lines), joined into a single overall contour $C$ (solid line).

Another way to think about this is to consider that one can construct any contour such as those described in the statement of Theorem 1 by joining together a finite number of contours, each one of which touches the unit circle at only one singular point, as is illustrated for the case of three contours in Figure 4. Thus we see that the enormous freedom to deform integration contours within the open unit disk without changing the value of the integrals, which is given to us by the Cauchy-Goursat theorem in its usual form, can be used to reduce a generic closed contour, that touches $N$ isolated integrable singular points on the unit circle, to a set of $N$ closed contours, each one of which touches the unit circle at only one such singular point. This effectively reduces the proof for the large and more complex contour to that for the simple contour with only one singularity.

In addition to this, using the Cauchy-Goursat theorem in its usual form, we may also deform any contour so that it morphs into one that touches the unit circle at any points on that circle where $w(z)$ is analytic. In other words, the integration contour may also run along any parts of the unit circle on which $w(z)$ has no singularities at all. This completes, therefore, the proof of Theorem 1.

In this section we have established that the extended version of the Cauchy-Goursat theorem, allowing for the presence of a finite number of isolated integrable singularities on the integration contour, holds for all inner analytic functions within the unit disk. In Section 5 we will generalize that result to all complex analytic functions, anywhere on the complex plane, using conformal transformations. Therefore, before we discuss the generalization of the theorem we must discuss these conformal transformations.