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

Cauchy TheoryComplex Analysis, Algebra & Topologyfree

One formula, three consequences — including why a holomorphic function is automatically infinitely differentiable.

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.