Using the fact that integral of dz/z=2*pi*i if C is a simple closed curve surrou

Question

Using the fact that integral of dz/z=2*pi*i if C is a simple closed curve surrounding the origin, use a change of variables to prove that integral of dz / (z - z knot) = 2*pi*i if the point z knot is on the contour C or in the interior of the contour C, and integral of dz/(z-z knot)= 0 if z knot is a point in the exterior of the contour C. Now, consider the integral dz/(z^2 + 1) where C: absolute value of z= 2. Use a partial fraction decomposition of 1/(z^2 +1) to evaluate the integral. Explain why it is not possible to use logarithms or the arctangent functions to evaluate the integral, and how this relates to the Cauchy Goursat Theorem.

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