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My definition of the theorem is stated in this section of the Wikipedia article.

My question is why point $a$ must be within the interior of $D$, and can't it also be in $U$ (where I mean the part of $U$ but not $D$ the domain where $f$ is analytic)(I think $D$ is smaller since $D$ is any simple closed contour lying entirely within $U$)?

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    @Victor: I suggest you brush up on some of the prereqs of studying functions of$a$complex variable before trying to understand this theorem...2011-07-17

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If $a$ is not in $D$, then $f(z)/(z-a)$ is holomorphic in $D$. So the integral around the boundary of $D$ (the contour) is equal $0$ by Cauchy's Theorem.