Cauchy integral formula for not necessarily star-shaped regions

Question

Let $f(z)$ be analytic on a region $R$. Let $u, w\in R$. Suppose $C$ is a simple closed curve containing $u, w$. Is it true that $$f(u)-f(w)=\frac{1}{2\pi i}\int_C\frac{f(z)}{z-u}-\frac{f(z)}{z-w}\,dz\,?$$

If $R$ is star-shaped, then the result follows directly from the Cauchy integral formula. But I only learned the Cauchy integral formula for star-shaped regions. Are there other subtleties to be dealt with? What if $R$ is not star-shaped?

Any help is much appreciated!

Answer 1

Cauchy's Theorem: Suppose that $R$ is an open set in $ \mathbb C$ and that $f:R \to \mathbb C$ is holomorphic. Furthermore let $c:[a,b] \to R$ a closed and piece-wise $C^1$ curve with $ ind_c(v)=0$ for all $v$ not in $R$, then

$$f(z) \cdot ind_c(z)=\frac{1}{2\pi i}\int_C\frac{f(w)}{w-z}dw$$

for all $z \in R \setminus c[a,b]$.

$ind_c( \cdot)$ denotes the winding number.