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Branch cut of the principle value log

  1. Feb 12, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the branch points of g(z) = log(z(z+1)/(z-1)) and defining a branch of g as the principle branch of the logarithm find the location of the branch cuts.


    2. Relevant equations



    3. The attempt at a solution

    Since [tex] g(z) = log(z) + log(z+1) - log(z-1) [/tex] the branch points are 0, 1, -1 and infinity.

    To find the branch cuts we need to find where the argument is [tex] arg(z) \in (-\pi,\pi] [/tex]

    putting each branch point into polar coordinates gives:
    [tex] z1 = r_1e^{i\theta_1} [/tex]
    [tex] z2 = r_2e^{i\theta_2} -1 [/tex]
    [tex] z3 = r_3e^{i\theta_3} +1[/tex]
    [tex] z = ln(\left|{\dfrac{r_1 r_2}{r_3}}\right|) +i(\theta_1 + \theta_2- \theta_3) [/tex]

    let [tex] c = ln(\left|{\dfrac{r_1 r_2}{r_3}}\right|) [/tex] and then:

    [tex] (-\infty, -1) \Rightarrow g(z) = c + \pi i [/tex]
    [tex] (-1, 0) \Rightarrow g(z) = c + 2\pi i [/tex]
    [tex] (0, 1) \Rightarrow g(z) = c + \pi i [/tex]
    [tex] (1, \infty) \Rightarrow g(z) = c [/tex]

    So this gives [tex] (0, 1) [/tex] as the point where the branch cut isn't defined, but the solution I was given says that [tex] (1, \infty) [/tex] is also not defined, so I'm not sure where I'm going wrong.
     
  2. jcsd
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