Cauchy Integral Theorem — Definition, Formula & Examples

The Cauchy Integral Theorem states that if a function is analytic (holomorphic) throughout a simply connected domain, then its integral around any closed contour in that domain equals zero.

Let $f$ be holomorphic on a simply connected open subset $D \subseteq \mathbb{C}$ , and let $\gamma$ be a piecewise smooth closed curve in $D$ . Then $\oint_\gamma f(z)\,dz = 0$ .

Key Formula

\oint_\gamma f(z)\,dz = 0

Where:

$f(z)$ = A function that is holomorphic (complex-differentiable) throughout the simply connected domain
$\gamma$ = A piecewise smooth closed contour contained in the domain
$dz$ = Complex line element along the contour

How It Works

To apply the theorem, verify two conditions: the function

f(z)

must be analytic at every point inside and on the contour, and the contour must lie within a simply connected region (no holes). If both conditions hold, the contour integral is immediately zero — no computation needed. When singularities exist inside the contour, the theorem does not apply directly, and you must instead use the Cauchy Integral Formula or the residue theorem.

Worked Example

Problem: Evaluate

\oint_{|z|=2} (z^2 + 3z)\,dz

Step 1: Identify the function and check analyticity. Here

f(z) = z^2 + 3z

, which is a polynomial. Polynomials are entire — analytic on all of

\mathbb{C}

Step 2: Check the domain. The circle

|z| = 2

encloses a simply connected region, and

f(z)

has no singularities anywhere.

Step 3: Apply the Cauchy Integral Theorem directly.

\oint_{|z|=2} (z^2 + 3z)\,dz = 0

Answer: The integral equals

0

Why It Matters

The Cauchy Integral Theorem is the foundation of most major results in complex analysis, including the Cauchy Integral Formula, Taylor and Laurent series, and the residue theorem. In physics and engineering, it simplifies the evaluation of real integrals that would otherwise be extremely difficult, particularly in electromagnetism and fluid dynamics.

Common Mistakes

Mistake: Applying the theorem when the function has a singularity inside the contour.

Correction: The theorem requires

f(z)

to be analytic everywhere inside and on the contour. If a singularity like a pole lies inside, use the residue theorem or the Cauchy Integral Formula instead.