Is it possible to have branch point in both the real and imaginary part of a complex function f(z)? An example might be
\begin{align} f(z) = u(\theta) + iv(\theta) \end{align}
where $u$ and $v$ are the real and imaginary components and are functions of $\theta$ only. A branch point exists at z = 0.
If so, is the contour integral around the branch point affected? The analysis I typically see shows
\begin{align} \lim_{\epsilon \rightarrow 0}\oint_C f(z=\epsilon e^{i\theta}) dz = 0 \end{align}
but in all those cases it is only the imaginary part that is multivalued.