Cauchy's theorem

Cauchy’s theorem (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

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

Closed-contour integrals of analytic functions vanish.

Why it’s central

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 $\Omega$). Same as “conservative field has path-independent line integrals” in 2D vector calc.
• The Cauchy integral formula (CIF), which expresses $f$ inside a contour from its boundary values.
• The deformation principle: contours can be deformed continuously through analytic regions without changing the integral.
• Everything that follows: 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 drops that and assumes only $f$ analytic, using a subdivision argument. The Green’s proof is the one we use; Goursat’s is the deeper result.

The two parallel stories united

Cauchy’s theorem is two applications of Green’s theorem combined into one complex identity. The C–R equations are exactly 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 $⟨u,-v⟩$ vanish by Green’s circulation and flux forms, and those are the real and imaginary parts of $\oint fdz$.

So the two 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.

What can go wrong: non-simply-connected domains

Cauchy’s theorem requires $\Omega$ to be simply connected. The classic 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, so the hypothesis fails and Cauchy’s theorem doesn’t apply.

This failure is the engine of the Residue theorem: closed-contour integrals around isolated singularities give $2\pi i$ times the 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.

Used constantly: shrink a complicated contour around a singularity to a tiny circle where the integral is easy, or push a real-axis integral up onto a large semicircle in the upper half-plane where decay takes 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 $2\pi i$, so $\oint dz/(z-z_0)=2\pi i\cdot n(C,z_0)$, where $n$ is the winding number.

This setup leads straight into the Cauchy integral formula and the Residue theorem.