Let $C$ and $D$ be two closed contours, $D$ lying completely within $C$, and let $a$ be a point between $C$ and $D$. Show that:
$$ f(a) = \frac{1}{2\pi}\int_{C}\frac{f(z)}{z-a}dz - \frac{1}{2\pi}\int_D \frac{f(z)}{z-a}dz. $$
I am confused because I thought this would be trivial, using Cauchy's Integral Theorem.
Hint: In order to use Cauchy's Theorem, you need a single closed curve. If you consider a path $E$ from a point in $C$ to a point in $D$, then you can "join" $C$ and $D$ into a single curve. (You might want to draw a picture.)
Of course, you'd like to be able to cancel any integration over $E$ - why can you do this?