Find g(Z) and show that is holomorphic Let be $\gamma$ the circumference of the center at the origin and radius 2, and we will consider the function $g:\mathbb{C}/\gamma \to\mathbb{C}$
$g(z)=\displaystyle\int_{\gamma}\frac{cos(s)}{z-is}ds$
I have doubts about how to calculate $g(z)$ for $ \left |z\right|\ne2 $ and then show that it is holomorphic in $\mathbb{C}$\ {$z:\left |z\right|=2$} then calculate $g´(z)$
Thanks in advance :)

Define $h(z)=g(iz)$.  Then $h$ is holomorphic with derivative $h'(z)=-1/(2\pi )\oint_{\gamma}\cos s/(z-s)^2\operatorname ds$, by Cauchy's differentiation formula.
But then $g'(z)=-ih'(-iz)$.
Meanwhile, for $|z|\gt2$, we have $g(z)=0$, by Cauchy's theorem.
