Cauchy integral formula

Let $f$ be a complex-valued holomorphic function on an open set $U\subseteq \mathbb{C}$, and let $\gamma$ be a positively oriented, simple closed contour contained entirely in $U$. If $a$ is a point inside $\gamma$, then for any non-negative integer $n$, we have:

with the important particular cases $n=0$:

and $n=1$:

This formula expresses the $n$-th derivative of a holomorphic function at a point $a$ as a contour integral around $a$. It highlights the remarkable fact that all derivatives of $f$ inside the region bounded by $\gamma$ are completely determined by the values of $f$ on the contour itself.

Consequences:

• Holomorphic functions are infinitely differentiable and analytic.
• Knowledge of a holomorphic function on a closed curve determines its behavior inside.
• The formula underpins many core results in complex analysis, including the development of power series and the residue theorem.

Proof

Here is the classic derivation showing that the Cauchy Integral Formula

is a corollary of the Cauchy--Goursat theorem.
Fix a point $a$ in the interior of $C$. Define

• Since $f$ is holomorphic on and inside $C$, $g$ is holomorphic there as well (the apparent pole at $z=a$ is removable, because $\underset{z\to a}{lim}g(z)=f^{\prime }(a)$).
• By Cauchy–Goursat,$${\oint }_{C}g(z)dz=0.$$

Write

Rearrange:

One shows by parameterizing $z=a+R{e}^{i\theta }$, $0\le \theta \le 2\pi$, that

So therefore