Cauchy's Integral Formula:
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The following theorem shows that the values of an analytic function $f$ interior to a simple closed contour $C$ are completely determined by the values of $f$ on $C$.

Cauchy's Integral Formula: Let $f$ be analytic in the simply connected domain $D$, and let $C$ be a simple positively oriented contour that lies in $D$. If $z_{0}$ is a point that lies interior to $C$, then MATH

Example: Find MATH.
Let MATH. Observe that $f(z)$ is analytic on and inside $\vert z \vert = 2$.
By applying Cauchy's integral formula, we get MATH Therefore, MATH

 
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