1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: A contour integral

  1. Nov 4, 2009 #1
    1. The problem statement, all variables and given/known data

    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].

    2. Relevant equations

    Cauchy's theorem, Residue theorem

    3. 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?
     
  2. jcsd
  3. Nov 4, 2009 #2
    Are you certain that a is not just an integer that can be 0, 1, or 2?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook