Let (C) be the positively oriented ellipse (3 x^{2}+y^{2}=9). Define [Fleft(z_{0} ight)=int_{C} frac{z^{2}+2 z}{z-z_{0}} d z] Find (F(2 i)) and (F(2)). [Sketch the ellipse in

Let \(C\) be the positively oriented ellipse \(3 x^{2}+y^{2}=9\). Define

\[F\left(z_{0}\right)=\int_{C} \frac{z^{2}+2 z}{z-z_{0}} d z\]

Find \(F(2 i)\) and \(F(2)\). [Sketch the ellipse in the complex plane. Use the Cauchy Integral Theorem with an appropriate \(f(z)\), or Cauchy's Theorem if \(z_{0}\) is outside the contour.]