Cauchy's integral formula, and what it gives you for free

If $f$ is holomorphic on and inside a simple closed contour $C$, then $\oint_C f = 0$. If $a$ is inside $C$,

$f(a) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z-a}\,dz.$

The value at an interior point is determined entirely by the boundary values.