For $z \in \mathbb{C} \backslash \gamma^*$, let $$f(z)=\int_{\gamma} \frac{dw}{w-z}.$$
I need to find the value of $f(z)$ when $\gamma(t)=(1+3\sin(t))e^{it}$, for $t \in [0,2\pi]$ (limaçon curve).
Firstly I tried to compute the integral by definition and I got nothing. Then I searched for an image of the curve and I see that if $z$ is the point marked below
Then Ind$(z)=2$, if it's in "the outter curve" it's $1$, and "out" it's $0$. How can I solve this, should I consider different positions $z$?