The bookUnit 2 · Cauchy Theory32 / 83

Cauchy's theorem and integral formula

Why this is asked: Pick the right tool: Cauchy's theorem (no singularities inside) vs the integral formula (one simple pole) vs residues (several). Simple connectedness is what makes ∮ = 0.

Cauchy's theorem and integral formula — the toolkit

The written notes for this page come with the Notes pack. The video, the visual and the practice below are free.

See pricing

See it move

Cauchy's integral formula, and what it gives you for freeinteractivefree

Step 1 / 4The theorem

If $f$ is holomorphic on and inside a simple closed contour $C$ , then $\oint_{C} f = 0$ . If $a$ is inside $C$ ,

$f (a) = \frac{1}{2 π i} \oint_{C} \frac{f ( z )}{z - a} d z .$

The value at an interior point is determined entirely by the boundary values.

Next: Liouville, Morera, maximum modulus principle

Create a free account to keep your place and have this feed your study plan.

Open this in the full syllabus view · Unit 2