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Cauchy's Theorem

If u and v satisfy the Cauchy-Riemann equations inside and on the simple closed contour C, then the integral of f(z)=0

Now for example, if we have f(z)=1/(z+20) and our closed contour is a circle around the origin with radius=1. If I am understanding this correctly, we can say that the integral is equal to 0 since the 'bad point' of z=-20 is outside of the circle correct meaning that f is differentiable in and on |z|=1.

Does this sound correct?