## Complex analysis: simple closed curve

Using the fact that closed curve 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 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=|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 arctangent functions to evaluate the integral, and how this relates to the Cauchy Goursat Theorem.