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Branch cuts

  1. Sep 28, 2008 #1
    I have a question with regards to branch cuts:

    Say I have a function f(z) = log(z2 - 1). Why is a simple branch cut connecting z = -1 and z = +1 not sufficient to define an analytic function? On the other hand, why is it sufficient for the function f(z) = (z^2 - 1)^(1/2) ?

    This is in the complex plane where z = a + ib.
     
  2. jcsd
  3. Sep 28, 2008 #2

    olgranpappy

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    perhaps the point at infinity?
     
  4. Sep 28, 2008 #3
    ^^ what do you mean by this?
     
  5. Sep 29, 2008 #4

    olgranpappy

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    Don't worry about.

    How about just trying to write
    [tex]
    z=1+r_1e^{i\theta_1}
    [/tex]
    and
    [tex]
    z=-1+r_2e^{i\theta_2}
    [/tex]
    and investigate how the function
    [tex]
    \log(z^2-1)=\log(z-1)+\log(z+1)=\log(r_1r_2)+i(\theta_1+\theta_2)
    [/tex]
    behaves as you go around the perspective "branch cut" that you mentioned... try it.
     
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