[Math] How to use Cauchy integral formula for this integral $g(z)=\int_{C}\frac{s^2+s+1}{s-z}ds$

complex-analysisintegration

Let $C$ be the ellipse $9x^2+4y^2=36$ traversed once in the counterclockwise direction. Define the function $g$ by $$g(z)=\int_{C}\frac{s^2+s+1}{s-z}ds.$$

Find $g(4i)$.

Well I know I must find $g(z)$ (that is the integral) before computing $g(4i)$, so I decided to use Cauchy's integral formula $f(z_{0})=\frac{1}{2\pi i}\int_C\frac{f(z)}{z-z_{0}}dz$. This put me into trouble, because I do not how to start. Please i need a hint.

Thanks.

Best Answer

Hint:

  1. Carefully sketch the curve $C$.
  2. Use some theorem of Cauchy.

It might also be interesting to look at other points than $z=4i$, e.g. $z=0$...

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