Laurent Expansion of Function Defined as Integral

Question

Let $z \in \mathbb{C} \setminus [0,1] $ and define $f(z) = \int^{1}_{0} \frac{x}{x-z} dx$. Find the Laurent Expansion of $f(z)$ in the annulus $A= [{z: |z|>1}]$, and determine its region of convergence and state the explicit form for $f(z)$.

Any hints would help.

Answer 1

Use the expansion $$ \frac{x}{x-z} = -\sum_{n=1}^\infty \left(\frac{x}{z}\right)^n $$ valid for $|x|<|z|$ and then integrate term by term...