Cauchy's Theorem and Applications

Cauchy's theorem, together with the Cauchy integral formula, provides the foundation for the analytic theory of holomorphic functions. Here we develop the deeper consequences and applications.

Taylor Series Representation

Definition4.1Analytic function

A function $f: D \to \mathbb{C}$ is analytic (or complex-analytic) at $z_0 \in D$ if there exists $r > 0$ and coefficients $\{a_n\}_{n=0}^\infty \subset \mathbb{C}$ such that

$f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n$

for all $z$ with $|z - z_0| < r$ . The function $f$ is analytic on $D$ if it is analytic at every point of $D$ .

Theorem4.1Holomorphic implies analytic

Let $f$ be holomorphic on a domain $D$ and let $z_0 \in D$ . If $d = \text{dist}(z_0, \partial D)$ , then $f$ has a Taylor series

$f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n$

converging for $|z - z_0| < d$ . The radius of convergence is at least $d$ and equals the distance from $z_0$ to the nearest singularity.

RemarkHolomorphic = Analytic

In complex analysis, the notions of holomorphic and analytic are equivalent. This is dramatically different from real analysis, where $C^\infty$ functions need not be real-analytic (e.g., $e^{-1/x^2}$ ). The equivalence is a consequence of the Cauchy integral formula.

Zeros of Analytic Functions

Definition4.2Order of a zero

If $f$ is holomorphic at $z_0$ and $f(z_0) = 0$ , the order (or multiplicity) of the zero at $z_0$ is the smallest $m \geq 1$ such that $f^{(m)}(z_0) \neq 0$ . Equivalently, $f(z) = (z - z_0)^m g(z)$ where $g$ is holomorphic and $g(z_0) \neq 0$ .

Theorem4.2Identity theorem

If $f$ and $g$ are holomorphic on a connected domain $D$ and $f = g$ on a set with an accumulation point in $D$ , then $f = g$ on all of $D$ .

ExampleApplication of the identity theorem

If $f$ is holomorphic on $\mathbb{C}$ and $f(1/n) = 1/n^2$ for all $n \in \mathbb{N}$ , then $f(z) = z^2$ on all of $\mathbb{C}$ . The set $\{1/n\}$ has accumulation point $0 \in \mathbb{C}$ , and $f(z) - z^2 = 0$ on this set, so by the identity theorem $f \equiv z^2$ .

Maximum Modulus Principle

Theorem4.3Maximum modulus principle

If $f$ is holomorphic and non-constant on a domain $D$ , then $|f|$ has no local maximum in $D$ . Equivalently, if $D$ is bounded and $f$ is continuous on $\overline{D}$ , then

$\max_{z \in \overline{D}} |f(z)| = \max_{z \in \partial D} |f(z)|.$

ExampleMaximum on the boundary

For $f(z) = z^2 + 1$ on the closed disk $|z| \leq 1$ , the maximum of $|f|$ occurs on $|z| = 1$ . At $z = \pm i$ : $|f(\pm i)| = |0| = 0$ . At $z = \pm 1$ : $|f(\pm 1)| = 2$ . At $z = e^{i\theta}$ : $|f(e^{i\theta})|^2 = |e^{2i\theta} + 1|^2 = 2 + 2\cos 2\theta$ , maximized at $\theta = 0$ , giving $|f| = 2$ .

RemarkMinimum modulus principle

If $f$ is holomorphic, non-constant, and nonvanishing on $D$ , then $|f|$ also has no local minimum in $D$ (apply the maximum principle to $1/f$ ). However, if $f$ has zeros, local minima of $|f|$ can occur at the zeros.