Cauchy integral formula

Statement

Let $D\subseteq\mathbb C$ be open, let $f$ be holomorphic on $D$ , and let $\gamma$ be a closed piecewise $C^1$ contour whose image lies in $D$ and whose index vanishes outside $D$ . For every $a\in D\setminus\gamma$ ,

\operatorname{Ind}(\gamma,a)f(a) =\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{z-a}\,dz.

Derivatives

Differentiation under the integral gives, for $n\ge0$ ,

\operatorname{Ind}(\gamma,a)f^{(n)}(a) =\frac{n!}{2\pi i}\int_\gamma\frac{f(z)}{(z-a)^{n+1}}\,dz.

For a positively oriented circle contained with its interior in $D$ , the index is $1$ inside. The derivative formula proves that holomorphic functions are infinitely differentiable and supplies Cauchy estimates.

Consequences

The formula is the main local engine behind analyticity, the maximum modulus principle, and Liouville's theorem.

References

John B. Conway, Functions of One Complex Variable I, 2nd ed., Springer, 1978. Publisher record. Relevant: Chapter IV, §5.