Evaluation of Spence's function.

Question

Spence's function is defined as $${\rm Li}_2 (z)=- \int_0^z \frac{\ln(1-u)}{u} \, du $$ where $$z \in {\mathbb C} \setminus [1, \infty )$$ For $|z|<1 $ $${\rm Li}_2 (z)= \sum_1^ \infty \frac{ z^k }{ k^2 } $$ Here the sum is over $k$. All this information is from Wiki. Can someone help me to find out how to evaluate the integral?

Answer 1

Hint: Expand $\ln(1-u)$ into its Mercator series, and then reverse the order of summation and integration.