Cauchy's theorem

Cauchy’s theorem (also called the Cauchy–Goursat theorem) is the central result of complex integration:

If $f$ is analytic on a simply connected open set $\Omega$, and $C$ is a closed contour in $\Omega$, then

${\oint }_{C}f(z)dz=0.$

That is, closed-contour integrals of analytic functions vanish.

Why this is the central theorem

Cauchy’s theorem implies:

• Path independence. For analytic $f$, ${\int }_A^{B}f(z)dz$ depends only on the endpoints — not on the path (within $\Omega$). The 2D vector-calculus analog of “conservative field has path-independent line integrals.”
• The Cauchy integral formula (CIF), which expresses $f$ inside a contour from its boundary values.
• The deformation principle: contours can be continuously deformed through analytic regions without changing the integral.
• Everything that follows in complex analysis: Taylor series of analytic functions, Liouville’s theorem, the Fundamental theorem of algebra, the Residue theorem.

Sketch via Green’s theorem

Writing $f=u+iv$ and $dz=dx+idy$:

${\oint }_{C}fdz={\oint }_C(u+iv)(dx+idy)={\oint }_C(udx-vdy)+i{\oint }_C(vdx+udy).$

Apply Green’s theorem to each piece. For the real part with $P=u,Q=-v$:

$\oint udx-vdy={\iint }_D(Q_x-P_y)dA={\iint }_D(-v_x-u_y)dA.$

By Cauchy-Riemann equations, $-v_x-u_y=u_y-u_y=0$ (using $u_y=-v_x$). So the real part vanishes.

For the imaginary part with $P=v,Q=u$:

$\oint vdx+udy={\iint }_D(u_x-v_y)dA=0$

by the other C–R equation $u_x=v_y$.

Both pieces vanish. ${\oint }_{C}fdz=0$. ∎

(Caveat: the Green’s-theorem proof needs $f$ to be $C^1$. Goursat’s original proof avoids the $C^1$ assumption — only $f$ analytic — using a clever subdivision argument. For practical purposes the Green’s proof is what we use; the deeper result is good for theoretical completeness.)

The two parallel stories united

The proof exposes the deep connection: Cauchy’s theorem is two applications of Green’s theorem combined into one complex identity. The C–R equations are precisely the conditions for the vector field $⟨u,-v⟩$ to be both conservative (zero curl) and source-free (zero divergence). When both hold, the circulation and flux of $⟨u,-v⟩$ both vanish by Green’s circulation and flux forms — and these are exactly the real and imaginary parts of $\oint fdz$.

So the “two parallel stories” of vector calculus — circulation/curl and flux/divergence — collapse into one in complex analysis. Complex analysis is the 2D vector calculus of doubly-special fields, in disguise.

What can go wrong: non-simply-connected domains

Cauchy’s theorem requires $\Omega$ to be simply connected. The classical failure: $f(z)=1/z$ is analytic on $\mathbb{C}\setminus \{0\}$, but ${\oint }_{|z|=1}dz/z=2\pi i\ne 0$.

The domain $\mathbb{C}\setminus \{0\}$ isn’t simply connected — the unit circle can’t be contracted to a point without crossing the missing origin. The hypothesis fails, so Cauchy’s theorem doesn’t apply.

In a deeper sense, this failure is the engine of the Residue theorem: closed-contour integrals around isolated singularities give $2\pi i$ times specific quantities (residues), instead of $0$.

Deformation principle

If $f$ is analytic on a domain (possibly with holes — i.e., not simply connected — as long as $f$ stays analytic everywhere in the domain), and we deform a closed contour continuously through the domain, the integral doesn’t change.

Proof: the difference between the two contours bounds a region where $f$ is analytic; Cauchy’s theorem on that region gives zero net contribution.

The deformation principle is invoked constantly: shrink a complicated contour around a singularity to a tiny circle around the singularity (where the integral is easy), or deform a real-axis integral up onto a large semicircle in the upper half-plane (where decay properties take over).

Applied to the fundamental integral $\oint dz/(z-z_0)$

Deform any closed contour $C$ enclosing $z_0$ once (counterclockwise) to a tiny circle of radius $\epsilon$ around $z_0$. Parameterize $z=z_0+\epsilon e^{it}$, $0\le t\le 2\pi$, $dz=i\epsilon e^{it}dt$, $z-z_0=\epsilon e^{it}$.

$\oint \frac{dz}{z-z_0}={\int }_0^{2\pi }\frac{i\epsilon e^{it}}{\epsilon e^{it}}dt={\int }_0^{2\pi }idt=2\pi i.$

So $\oint dz/(z-z_0)=2\pi i$ for any positively oriented closed contour enclosing $z_0$ once.

If $C$ doesn’t enclose $z_0$: $1/(z-z_0)$ is analytic on a neighborhood of $C$ and its interior, so $\oint =0$ by Cauchy’s theorem.

If $C$ winds around $z_0$ multiple times: each loop contributes $2\pi i$. The general formula: $\oint dz/(z-z_0)=2\pi i\cdot n(C,z_0)$, where $n$ is the winding number.

This setup leads directly to the Cauchy integral formula and ultimately to the Residue theorem.