I'm trying calcualte this integral $\int_C \frac{z^3+3}{z(z-i)}dz$ when the path $C$ is something similar to $$C=\gamma_1+\gamma_2$$, where $\gamma_1:|z-i|=1$, $\gamma_2:|z+i|=1.$
My solutions is choose $f(z)$ as $f(z) = \frac{z^3 + 3}{z}$ but $f$ is not analytic on the path, I don't know how choose the function because $i$ is inside of the curve $C$ then I think that my unique choose of $f$ it's the one I gave before.
Can you give me a hint on how to proceed?