Cauchy integral formula

Statement

For a circle with respect to any point in the interior

Suppose $U$ is a domain in $\mathbb{C}$ and $f:U\to \mathbb{C}$ is a holomorphic function. The Cauchy integral formula states that if $z_0\in U$ is such that the disc of radius $r$ about $z_0$ lies completely inside $U$, and if $z$ in such that $|z-z_0|<r$, we have:

$f(z)=\frac{1}{2\pi i}{\oint }_{\partial D}\frac{f(\xi )}{\xi -z}d\xi$

In other words, we can determine the value of $f$ at the center, by knowing its value on the boundary.

For any simple closed curve and any point in the interior

This is the same integral formula, this time applied to any simple closed curve and any point in the interior.

Suppose $U$ is a domain in $\mathbb{C}$ and $f:U\to \mathbb{C}$ is a holomorphic function. The Cauchy integral formula states that if $\gamma$ is a smooth simple closed curve lying completely inside $U$, and $z_0$ is in the component of the complement of $\gamma$ that lies completely inside $U$, then we have:

$f(z_0)=\frac{1}{2\pi i}{\oint }_{\partial D}\frac{f(z)}{z-z_0}dz$

In other words, the curve doesn't need to be a circle and the point can be anywhere inside.

For any union of piecewise smooth curves and any point

Suppose $U$ is an open subset of $\mathbb{C}$ and $f:U\to \mathbb{C}$ is a holomorphic function.

Suppose $c$ is a sum of oriented piecewise smooth loops $c_1,c_2,\dots ,c_r$, each completely inside $U$. In other words, when we use the symbol ${\int }_{c}g(z)dz$, we actually mean $\sum _1^r{\int }_{c_r}g(z)dz$.

Denote by $n(c;z_0)$ the sum of the winding numbers of the $c_k$ about $z_0$. Suppose $c$ is zero-homologous, i.e. $n(c;z)=0$ for $z\in \mathbb{C}\setminus U$. Then we have, for any $z_0\in U$:

$n(c;z_0)f(z_0)=\frac{1}{2\pi i}{\int }_c\frac{f(z)}{z-z_0}dz$

Related facts

• Cauchy integral formula for derivatives