quasar_4
Nov4-09, 04:18 PM
1. The problem statement, all variables and given/known data
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 .
2. Relevant equations
Cauchy's theorem, Residue theorem
3. 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?
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 .
2. Relevant equations
Cauchy's theorem, Residue theorem
3. 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?