Theorem (Cauchy's Integral Formula)

Letbe a simply connected region, letbe a simple closed contour inandbe a function analytic onThenfor any pointinside

Proof: Consider the integral

By the shrinking contour theorem we can replaceby any circleof radiusand centre lying insideto obtain(1)

Let

using the parametrization

Then

is continuous atso for eachthere issuch that

Now chooseto be any positive number such thatthen

forin

Hencesinceis the length of Sinceis arbitrarily small

then from (1)

Cauchy's integral formula can be used in a variety of ways, of which more later.