How to Apply Contour Integration to Complex Fractional Powers?

[quasar_4]

Homework Statement

Compute $$\int_{\alpha}^{\beta}{\left(\frac{\beta - x}{x-\alpha}\right)^{a-1} \frac{dx}{x}}$$ where $$0 \leq a \leq 2$$ and $$0 \leq \alpha \leq \beta$$.

Homework Equations

Cauchy's theorem, Residue theorem

The Attempt at a Solution

I'm confused about setting this up. If $$a \neq 1$$, then the function is multi-valued and we'd need a branch cut - but I don't understand where to put this branch cut. Also, what about the case where $$a = 1$$? Does this mean that there is more than one answer, depending on what a is?

Also, I can see that there is a simple pole at x=0 and some type of singularity at $$x=\alpha$$ (a pole of order a-1??) So, can I just use the Residue theorem once I figure out what contour to choose?

 

[n!kofeyn]

Are you certain that a is not just an integer that can be 0, 1, or 2?