I have a contour integral of a function of the form $(z^6-P)^\alpha z^\beta$ Here $\alpha\in R$, $\beta\in N$ and $P$ is some constant. I therefore have branch points at the sixth roots of $P$.
The contour I have is any contour that includes the two real branch points $\pm P^{1/6}$ but not any of the other roots of $(z^6-P)$. Here's the one I chose (figure from Arfken and Weber for a similar problem) although I am open to better suggestions if there are any.
The two circles enclose the points $\pm P^{1/6}$ and the line joining them is a branch cut. My questions is, how should my function be written as to get a single valued function in this contour? I assume that once I have a single valued function, this integral can be easily done? Thank you.