KindiakMath

Cauchy-Goursat Theorem

Previously, we have seen that if f has an antiderivative F on a domain D, then for any closed smooth contour \gamma \subseteq D,

\displaystyle \oint_{\gamma} f(z)\, \mathrm dz = 0,

and vice versa. When does this hold?

This post will take heavy reference from this document.

Theorem 1 (Cauchy-Goursat Theorem). Let U \subseteq \mathbb C be open and f : \mathbb C \to \mathbb C. If f is holomorphic on U, then for any bounded domain D with \bar D \subseteq U,

\displaystyle \oint_{\partial D} f(z)\, \mathrm dz = 0.

Lemma 1. Theorem 1 holds whenever \bar D is a closed rectangle.

Proof. Firstly, since f is holomorphic on U, it is continuous on \bar D, so that there exists \alpha > 0 such that

\displaystyle \left| \oint_{\partial D} f(z)\, \mathrm dz \right| \leq \alpha.

Suppose for a contradiction that equality holds. Then divide D into 4 equal sub-rectangles. By the triangle inequality and the pigeonhole principle, at least one of these rectangles D^{(1)} with \| \partial D^{(1)} \| = \| \partial D \| / 2 must yield

\displaystyle \left| \int_{\partial D^{(1)}} f(z)\, \mathrm dz \right| \geq \frac \alpha 4.

Inductively obtain a decreasing sequence D =: D^{(0)} \supseteq D^{(1)} \supseteq D^{(2)} \supseteq \dots of rectangles such that

\displaystyle \left| \oint_{\partial D^{(n)}} f(z)\, \mathrm dz \right| \geq \frac \alpha{4^n}.

Each rectangle D^{(m)} for each m > 0 has a center z_m. Since the sequence \{z_m\} is Cauchy, it converges to some z_0 \in \bar D. Define the map E by

E(z) := f(z) - (f(z_0) + f'(z_0) \cdot (z - z_0)).

Fix \epsilon > 0. Since f is holomorphic at z_0, there exists \delta > 0 such that

0 < |z - z_0| < \delta \quad \Rightarrow \quad |E(z)| < \epsilon.

Since z_m \to z_0, find m such that D^{(m)} \subseteq B(z_0, \delta). Since the map

f(z) - E(z) = f(z_0) + f'(z_0) \cdot (z-z_0)

has an anti-derivative f(z_0) \cdot (z-z_0) + f'(z_0) \cdot (z-z_0)^2/2,

\displaystyle \oint_{ \partial D^{(m)} } E(z)\, \mathrm dz = \oint_{ \partial D^{(m)} } f(z)\, \mathrm dz.

By construction, \|\partial D^{(m)}\| = \| \partial D \|/2^m. By the ML-inequality,

\displaystyle \frac{\alpha}{4^m} \leq \left| \oint_{ \partial D^{(m)} } E(z)\, \mathrm dz \right| \leq \epsilon \cdot \frac{\| \partial D \| }{ 2^m }.

Thus, the inequality yields a contradiction if

\displaystyle \frac{\alpha}{2^m \cdot \| \partial D \| } > \epsilon.

Therefore, set \epsilon = 1 and take m to be sufficient large to produce the desired contradiction. Therefore,

\displaystyle \left| \oint_{\partial D} f(z)\, \mathrm dz \right| = 0\quad \Rightarrow \quad \oint_{\partial D} f(z)\, \mathrm dz = 0.

Lemma 2. Suppose that U \subseteq \mathbb C is simply-connected and f is holomorphic on U. Then f has an anti-derivative F on U. Equivalently,

\displaystyle \oint_{\Gamma} f(z)\, \mathrm dz

for any closed contour \Gamma \subseteq D.

Proof. Fix z_0 \in U. For any z \in U, let \Gamma_1, \Gamma_2 be two paths from z_0 to z comprised only of finitely many horizontal and vertical paths. Then it is not hard to prove that there exist disjoint open rectangles D_m such that \bar D_m \subseteq U and

\displaystyle \partial \left( \bigsqcup_{m=1} D_m \right) = \Gamma_1 + (-\Gamma_2).

Therefore,

\displaystyle \begin{aligned} \oint_{\Gamma_1} f(z)\, \mathrm dz - \oint_{\Gamma_2} f(z)\, \mathrm dz &= \oint_{\Gamma_1 + (-\Gamma_2)} f(z)\, \mathrm d z \\ &= \sum_{m=1}^n \oint_{\partial D_m} f(z)\, \mathrm dz \\ &= \sum_{m=1}^n 0 = 0. \end{aligned}

Thus, the map F : \mathbb C \to \mathbb C defined by

\displaystyle F(z) = \oint_{\Gamma} f(z)\, \mathrm dz,

where \Gamma is any path from z_0 to z comprised only of finitely many horizontal and vertical paths is well-defined. To establish the antiderivative property, work with the path \Gamma = \gamma_1 + \gamma_2 where each path \gamma_1 = r_1([0, 1]) and \gamma_2 = r_2([0, 1]) is (without loss of generality) parameterised by

r_1(t) = z_0 + t \cdot \mathrm{Re}(z - z_0),\quad r_2(t) = r_1(1) + it \cdot \mathrm{Im}(z - z_0).

Of course, we verify that

\begin{aligned} r_2(1) &= r_1(1) + i \cdot \mathrm{Im}(z - z_0) \\ &= z_0 + \mathrm{Re}(z - z_0) + i \cdot \mathrm{Im}(z - z_0) \\ &= z_0 + (z - z_0) = z. \end{aligned}

so that \Gamma is indeed a path from z_0 to z. Taking |z - z_0| sufficiently small and employing the continuity of f at z_0, the ML-inequality yields

\displaystyle |F(z) - F(z_0) - f(z_0) \cdot (z-z_0)| = \left| \oint_{\Gamma} (f(z) - f(z_0)) \right| \leq \epsilon \cdot |z - z_0|,

establishing differentiability at z_0. Since U is open and z_0 \in U is arbitrary, F is holomorphic on U with F' = f.

We now handle the general case of Theorem 1.

Proof of Theorem 1. First suppose the special case that U is simply connected. Then \partial D \subseteq U is a closed contour, so that by Lemma 2,

\displaystyle \oint_{\partial D} f(z)\, \mathrm dz = 0.

For the general case, find a finite number of line segments \gamma_m with end-points in the smooth parts of \partial D so that D_0 := D \backslash \bigcup_{m=1}^n \gamma_m is simply connected. Intuitively, we “chop” U up into a collection of simply connected “islands”. For each \epsilon > 0, define

\displaystyle \gamma_{m,\epsilon} := \left\{z \in \mathbb C : \sup_{w \in \gamma_m} | z - w | \leq \epsilon \right\},\quad D_\epsilon := D \backslash \bigcup_{m=1}^n \gamma_{m, \epsilon} \subseteq D_0.

For sufficiently small \epsilon, \bar D_\epsilon \subseteq U for some simply connected neighborhood U, so that

\displaystyle \oint_{\partial D_\epsilon} f(z)\, \mathrm dz = 0.

Then using arguments involving uniform continuity, the ML-inequality, and the definition of the contour integral, take \epsilon \to 0^+ to obtain

\displaystyle \oint_{\partial D} f(z)\, \mathrm dz = \lim_{\epsilon \to 0^+} \oint_{\partial D_\epsilon} f(z)\, \mathrm dz = 0.

We took great pains to prove the Cauchy-Goursat theorem so that we can prove an even more incredible result—the Cauchy integral formula. This we will do the next time, as well as establish the many powerful properties of holomorphic functions that set it apart in the complex analytic setting from usual differentiability in the real-analytic setting.

—Joel Kindiak, 18 Aug 25, 1756H

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