Cauchy Integral Formula

The Cauchy integral formula expresses the values of a holomorphic function inside a closed contour in terms of an integral over the contour. It is arguably the single most important result in complex analysis, from which virtually all other major theorems follow.

The Cauchy Integral Formula

Definition3.7Cauchy kernel

The Cauchy kernel is the function

$C(z, w) = \frac{1}{2\pi i} \cdot \frac{1}{w - z}$

which, for fixed $z$ , is a holomorphic function of $w$ on $\mathbb{C} \setminus \{z\}$ . It serves as the reproducing kernel for holomorphic functions.

Theorem3.6Cauchy integral formula

Let $f$ be holomorphic inside and on a simple closed contour $\gamma$ (positively oriented). Then for every $z$ inside $\gamma$ :

$f(z) = \frac{1}{2\pi i} \oint_\gamma \frac{f(w)}{w - z}\,dw.$

RemarkRemarkable consequences

The Cauchy integral formula is remarkable for several reasons:

It recovers the interior values of $f$ from its boundary values alone.
It immediately implies that holomorphic functions are infinitely differentiable.
It provides integral representations that are powerful tools for estimation.
It shows that complex differentiability (a local condition) has global consequences.

Derivatives via the Cauchy Formula

Theorem3.7Cauchy integral formula for derivatives

Under the same hypotheses, for every $n \geq 0$ :

$f^{(n)}(z) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(w)}{(w-z)^{n+1}}\,dw.$

In particular, holomorphic functions are infinitely differentiable ( $C^\infty$ ), and all derivatives are again holomorphic.

ExampleComputing derivatives via integration

To find $f''(0)$ where $f(z) = e^z$ using the Cauchy formula with $\gamma = \{|z| = 1\}$ :

$f''(0) = \frac{2!}{2\pi i} \oint_{|z|=1} \frac{e^z}{z^3}\,dz.$

Since $f''(0) = e^0 = 1$ , we deduce $\oint_{|z|=1} e^z/z^3\,dz = \pi i$ .

Cauchy's Inequality and Liouville's Theorem

Theorem3.8Cauchy's inequality

If $f$ is holomorphic on the closed disk $\overline{D}(z_0, R)$ and $|f(z)| \leq M$ for $|z - z_0| = R$ , then

$|f^{(n)}(z_0)| \leq \frac{n! M}{R^n}.$

ExampleApplication of Cauchy's inequality

Let $f$ be an entire function with $|f(z)| \leq A + B|z|^k$ for some constants $A, B$ and positive integer $k$ . Applying Cauchy's inequality on a disk of radius $R$ for $n > k$ :

$|f^{(n)}(0)| \leq \frac{n!(A + BR^k)}{R^n} \to 0 \quad \text{as } R \to \infty.$

Therefore $f^{(n)}(0) = 0$ for all $n > k$ , so $f$ is a polynomial of degree at most $k$ .

Mean Value Property

Definition3.8Mean value property

A continuous function $f$ on a domain $D$ satisfies the mean value property if for every closed disk $\overline{D}(z_0, r) \subset D$ :

$f(z_0) = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + re^{i\theta})\,d\theta.$

That is, $f(z_0)$ equals its average over every circle centered at $z_0$ .

RemarkHolomorphic functions and the mean value property

The Cauchy integral formula immediately implies that every holomorphic function satisfies the mean value property: setting $z = z_0$ and $w = z_0 + re^{i\theta}$ ,

$f(z_0) = \frac{1}{2\pi i} \oint_{|w-z_0|=r} \frac{f(w)}{w - z_0}\,dw = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + re^{i\theta})\,d\theta.$

The mean value property also characterizes harmonic functions. For holomorphic $f$ , both $\text{Re}(f)$ and $\text{Im}(f)$ are harmonic and satisfy this property.

ExampleMean value of $z^2$ on a circle

For $f(z) = z^2$ on the circle $|z - 1| = 2$ :

$\frac{1}{2\pi}\int_0^{2\pi} (1 + 2e^{i\theta})^2\,d\theta = \frac{1}{2\pi}\int_0^{2\pi} (1 + 4e^{i\theta} + 4e^{2i\theta})\,d\theta = 1$

since the integrals of $e^{in\theta}$ over $[0, 2\pi]$ vanish for $n \neq 0$ . Indeed $f(1) = 1$ .

Summary

RemarkCentral role of the Cauchy formula

The Cauchy integral formula is the engine that drives complex analysis. From it we derive:

Infinite differentiability of holomorphic functions
Power series representations (Taylor and Laurent)
Liouville's theorem and the Fundamental Theorem of Algebra
The maximum modulus principle
Residue calculus
Morera's theorem (converse of Cauchy's theorem)

Understanding this formula deeply is essential for all subsequent topics.