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: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted