I'm currently reviewing complex analysis and came across the following exercise.
Suppose that $f$ is holomorphic in the set $D_1(0)^C = \big\{z\;|\; |z|>1\big\}$, continuous up to $|z|=1$, and bounded as $z \rightarrow \infty$. Find an analog of the Cauchy integral formula expressing $f(x)$ for $z \in D_1(0)^C$ as a complex weighted average of the values of $f$ on the unit circle.
I'm having a difficult time figuring out how to get started with this exercise. My current thoughts are that we should use the Cauchy integral formula for a disc contained in $D_1(0)^C$ and somehow extend it so that it touches the unit disc at one point.
Any hints for how to get started would be greatly appreciated.
If $0<|z|<1$, let $g(z)=f\left(\frac1z\right)$. Since $g$ is bounded, by Riemann's theorem you can can extend it to an analytic function from $D_1(0)$ into $\Bbb C$, which can be extend to a continuous function form $\overline{D_1(0)}$ into $\Bbb C$. If $|c|>1$, then $\left|\frac1c\right|<1$ and, by Cauchy's theorem\begin{align}f(c)&=g\left(\frac1c\right)\\&=\frac1{2\pi i}\int_{|z|=1}\frac{g(z)}{z-1/c}\,\mathrm dz\\&=\frac1{2\pi i}\int_{|z|=1}\frac{f(1/z)}{z-1/c}\,\mathrm dz.\end{align}