Compute $\int_{\{|z|=4\}}\frac{z^2+2}{(z-3)^2}dz$

Question

I want to compute the value of the complex line integral $$\int_{\{|z|=4\}}\frac{z^2+2}{(z-3)^2}dz$$ I don't think I can apply Cauchy's integral theorem nor Cauchy's integral formula here. So far, I have tried parametrizing $\{|z|=4\}$ as $\gamma(\theta)=4e^{i\theta}$ and expanding the function I am integrating, however I don't think this is a good way to proceed. Can someone help me?

Answer 1

In fact, you can use Cauchy's integral formula for derivatives, with the holomorphic function $f:\mathbb{C}\to \mathbb{C}, f(z)=z^2+2$.

Then, for any $z_0\in \mathbb{C}$ with $|z_0|<4$, we get

$$\frac{1}{2\pi i}\int_{|z|=4}\frac{f(z)}{(z-z_0)^2}dz=f'(z_0)=2z_0.$$

Can you take it from here?

Answer 2

Use $\frac{f^{(n)}(z_0)}{n!}Ind_{\gamma}(z_0)=\frac{1}{2\pi i}\int_{\gamma}\frac{f(z)}{(z-z_0)^{n+1}}dz$.
(Ind = Winding number)