2.1Cauchy's Theorem in a Disk

Updated 15 Aug 2026Chapter (PDF)

[2.1.1]Lemma(Holomorphic Function on Open Disk Admits Primitive)#

Suppose $f$ is holomorphic on the open disk $\Omega$ . Then $f$ admits a primitive.

Proof.

Without loss of generality, let $\Omega$ be centered at the origin. Let $z \in \Omega$ be arbitrary, and consider a path $0 \to \Re(z) \to z$ in the orientation specified, and let it be $\gamma_z$ .

Then define $F$ by

F(z) = \int_{\gamma_z} f(w) \, \dd w.

We will prove that $F$ is holomorphic on $\Omega$ , and that $F'(z) = f(z)$ . Fixing $z \in \Omega$ , let $h \in \bC$ be such that $z + h \in \Omega$ (by virtue of $\Omega$ being an open disk). Then write

F(z + h) - F(z) = \int_{\gamma_{z + h}} f(w) \, \dd w - \int_{\gamma_z} f(w) \, \dd w.

We may draw out the contours $\gamma_{z + h}$ and $-\gamma_z$ , cancel out the contour from $0 \to \Re(z)$ , then draw an auxiliary triangle and yield the resultant contour $\gamma_{\text{res}}$ as a straight line connecting $z, z + h$ .

Thus

F(z + h) - F(z) = \int_{\gamma_{\text{res}}} f(w) \, \dd w.

Note that $f$ is continuous at $z$ , so we may write

f(w) - f(z) = \psi(w) \implies f(w) = f(z) + \psi(w),

where $\psi(z) \to 0$ as $w \to z$ . Thus we may write the integral as

F(z + h) - F(z) = f(z) \int_{\gamma_{\text{res}}} \dd w + \int_{\gamma_{\text{res}}} \psi(w) \, \dd w.

Obviously, the first integral has the primitive $w$ , and so the first term evaluates to $f(z) \times h$ . As for the second integral, we may use the ML inequality to yield

\abs{\int_{\gamma_{\text{res}}} \psi(w) \, \dd w} \le \sup_{w \in \gamma_{\text{res}}} |\psi(w)| \times |h|.

As $h \to 0$ , note that supremum goes to $0$ , and so we have

\lim_{h \to 0} \frac{F(z + h) - F(z)}{h} = F'(z) = f(z)

by simple rearrangement, completing the proof.

[2.1.2]Remark(Figures for Cauchy Theorem in a Disk)#

[2.1.3]Theorem(Cauchy's Theorem on a Disk)#

Suppose $f$ is holomorphic on some open disk $\Omega$ . Then, for any closed curve $\gamma \subseteq \Omega$ , we have that

\int_\gamma f(z) \, \dd z = 0.

Proof.

$f$ admits a primitive on $\Omega$ .

The idea that a holomorphic function on a disk attains a primitive is not unique to disks only. Indeed, the technique of defining our primitive by integrating over a piecewise-smooth path relies purely on our ability to systematically construct polygonal paths, consisting of finitely many horizontal and vertical segments, from a fixed base point $z_0$ to any point $z$ in the domain without exiting the region of holomorphicity. This geometric framework generalizes naturally to the class of toy contours—such as keyholes, sectors, and polygons—where the definition of the “interior” remains intuitive. Within these toy domains, Goursat's theorem for rectangles and triangles guarantees that integrals along any two such polygonal paths agree, yielding a well-defined, single-valued primitive

F(z) = \int_{\gamma_z} f(w) \, \dd w,

which immediately forces the integral over the closed boundary contour to vanish.

However, extending this primitive construction to arbitrary simple closed curves in the plane introduces deep topological difficulties. Defining the interior of a general closed curve rigorously is a non-trivial task, as the curve itself may be highly non-convex or geometrically intricate. Because of the complexity inherent in establishing that the interior of any simple closed curve is well-defined and simply connected, the general proof of Cauchy's theorem for all piecewise-smooth Jordan curves is deferred to Appendix B. There, we will use the topological properties of winding numbers and the Jordan Curve Theorem to establish the equivalence between holomorphic and topological simple connectivity, rigorously completing the generalization of Cauchy's theorem.