Cauchy's Integral Theorem — Mathematics Knowledge System

Overview

Cauchy's integral theorem (also called the Cauchy–Goursat theorem) states that if f is analytic throughout a simply connected domain D, then the contour integral of f over any closed contour C lying in D is zero: ∮_C f(z) dz = 0. This is the single most important theorem in complex analysis — it explains why contour integrals of analytic functions are path-independent within a simply connected region and it is the launching point for the Cauchy integral formula, Taylor series, and the residue theorem. The Goursat refinement shows the result holds even without assuming f' is continuous, using only analyticity itself. The theorem fails when the domain has 'holes' punctured by singularities, or when f is not analytic somewhere inside C.

Intuition

If f is analytic throughout a region with no holes, you can continuously deform any closed loop C down to a point without ever leaving the region or crossing a singularity, and the integral ∮_C f dz changes continuously (in fact stays constant) as you shrink the loop — because f is analytic, there's nothing 'inside' contributing extra area to the integral, unlike, say, integrating over a hole punched by a pole. Shrinking the loop to a point forces the integral to shrink to 0. The theorem fails precisely when the region has a hole (a singularity) that the loop cannot be shrunk past — which is exactly the setting where the residue theorem later measures what value the integral picks up.

Formal Definition

Definition

Let D be a simply connected open subset of ℂ and f analytic on D. For any closed, piecewise-smooth contour C contained in D:

\oint_C f(z)\,dz = 0

Cauchy's integral theorem

\int_{C_1} f(z)\,dz = \int_{C_2} f(z)\,dz

Consequence: path-independence between two curves C1, C2 with the same endpoints, homotopic within D

Path independence

Worked Examples

e^z is entire (analytic on all of ℂ), and ℂ is simply connected, so Cauchy's theorem applies directly.
$e^z \text{ analytic everywhere}$
By Cauchy's integral theorem, the integral over any closed contour vanishes.
$\oint_C e^z\,dz = 0$

Answer: ∮_C e^z dz = 0 for any closed contour C, since e^z is entire.

Practice Problems

Difficulty 4/10

Evaluate ∮_C (z^3 - 3z + 5) dz where C is any closed contour in ℂ.

Difficulty 5/10

Let C1 be the straight segment from 0 to 1+i, and C2 be the path from 0 to 1 to 1+i (two segments). Explain why ∫_{C1} z dz = ∫_{C2} z dz for f(z)=z.

Difficulty 6/10

Does Cauchy's theorem guarantee ∮_C 1/(z-3) dz = 0 for C the unit circle |z|=1?

Quiz

Summary

If f is analytic throughout a simply connected domain D, then ∮_C f(z) dz = 0 for every closed contour C in D.
This forces path-independence of contour integrals of analytic functions within simply connected regions.
The theorem fails when a singularity of f lies inside C — exactly the situation quantified later by the residue theorem.

References

WebsiteWikipedia — Cauchy's integral theorem