The question is, with an interval $I=[a,b]\subset\mathbb{R}$, as well as a function $f:I\to\mathbb{C}$ how to show the following function: $$F(z)=\int_a^b\frac{f(t)}{t-z}\,dt$$ is analytic on the domain $\mathbb{C}\setminus I$?
I just want to know the best procedure to do so. I guess I should use the definition of the complex derivative, but not sure if I need to solve the integral as a function of $t$ first.