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## Homework Statement

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

## Homework Equations

Cauchy's theorem, Residue theorem

## The Attempt at a Solution

I'm confused about setting this up. If [tex] a \neq 1 [/tex], 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 [tex] a = 1 [/tex]? 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 [tex] x=\alpha [/tex] (a pole of order a-1??) So, can I just use the Residue theorem once I figure out what contour to choose?