Give a brief description of what is meant by analytic continuation.

The dilogarithm function is defined by

$$\mathrm{Li}_{2}(z)=\sum_{n=1}^{\infty} \frac{z^{n}}{n^{2}}, \quad|z|<1$$

Let

$$f(z)=-\int_{C} \frac{1}{u} \ln (1-u) d u$$

where $C$ is a contour that runs from the origin to the point $z$. Show that $f(z)$ provides an analytic continuation of $\mathrm{Li}_{2}(z)$ and describe its domain of definition in the complex plane, given a suitable branch cut.