Cauchy's Theorem

5.1 Statement

Theorem 5.1 (Cauchy”s Theorem). If $f$ is analytic on a connected domain $D$ and $\gamma$ Is a simple closed contour in $D$Then

$\int_\gamma f(z)\, dz = 0$

Proof (for $f'$ continuous). By Green’s theorem in the plane, writing $f = u + iv$:

$\int_\gamma f\, dz = \int_\gamma (u\, dx - v\, dy) + i\int_\gamma (v\, dx + u\, dy)$

Applying Green’s theorem to each integral:

$= \iint_D (-v_x - u_y)\, dA + i\iint_D (u_x - v_y)\, dA = 0$

By the Cauchy-Riemann equations. $\blacksquare$

5.2 Connected Domains

A domain $D \subseteq \mathbb{C}$ is ** connected** if every simple closed contour in $D$ can Be continuously shrunk to a point within $D$.

Cauchy’s theorem may fail on multiply connected domains. For example, $\int_\gamma \frac{1}{z}\, dz = 2\pi i$ where $\gamma$ is the unit circle (traversing a region that Excludes the singularity at $z = 0$).

5.3 Path Independence

Corollary 5.2. If $f$ is analytic on a connected domain $D$Then the integral $\int_{z_0}^{z_1} f(z)\, dz$ is independent of the path from $z_0$ to $z_1$ in $D$.

5.4 Antiderivatives

Theorem 5.3. If $f$ is analytic on a connected domain $D$Then $f$ has an antiderivative $F$ in $D$ (i.e., $F'(z) = f(z)$), and

$\int_\gamma f(z)\, dz = F(z_1) - F(z_0)$

Where $z_0$ and $z_1$ are the endpoints of $\gamma$.

5.5 Cauchy’s Theorem for Multiply Connected Domains

Theorem 5.4. If $f$ is analytic on a domain $D$ containing simple closed contours $\gamma, \gamma_1, \ldots, \gamma_n$ where $\gamma_1, \ldots, \gamma_n$ Lie in the interior of $\gamma$ and the region between $\gamma$ and the $\gamma_k$ is contained in $D$ And all contours are positively oriented, then

$\int_\gamma f(z)\, dz = \sum_{k=1}^n \int_{\gamma_k} f(z)\, dz$

5.6 Deformation of Contours

Theorem 5.5 (Deformation of Contours). If $f$ is analytic on a domain containing two simple Closed contours $\gamma_1$ and $\gamma_2$ where one can be continuously deformed into the other Within the domain of analyticity of $f$Then

$\int_{\gamma_1} f(z)\, dz = \int_{\gamma_2} f(z)\, dz$

Proof. This follows directly from Theorem 5.4 applied to the region between $\gamma_1$ and $\gamma_2$. $\blacksquare$

Remark. This theorem is enormously useful: we can replace a complicated contour with a simpler one (a small circle around each singularity) without changing the value of the integral.