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Branch-cut singularity of a complex logarithm

  1. Mar 28, 2009 #1
    Hi!

    Does anyone know what a branch-cut singularity is? I have been trying to understand its importance in physics, but I got lost. I would guess that a singularity in physical context should mean that the value of a function should become very big near that singularity. But if we take complex logarithm, we can become big only in two cases, when the argument is either 0 or infinity.
    However, people choose the negative part of a real line in a complex plane as a branch cut for a complex logarithm, and say that this branch cut is a weak singulartiy compared to a simple pole. What do they mean by that?
     
  2. jcsd
  3. Apr 5, 2009 #2
    The problem with log z, is that the function is not single-valued if you allow paths which wind around the complex origin. If you approach a branch cut from different paths, you may get different limits. Towards a simple pole you get no limits at all. How they compare singularities may depend on how they effect integrals, etc.The question is a bit vague, so this is my best guess at what they mean.
     
  4. Apr 13, 2009 #3
    You have got to understand log(z) first: Plot the real and imaginary components of log(z)=ln|z|+arg z. You should get a funnel for the first and a cork-screw for the second. Take it slow first: Use ParametricPlot3D[{Re[z],Im[z],t}/.z->r Exp[it],{r,0,2},{t,-10,10}] in Mathematica to see the cork-screw. But that surface is not single valued right? Can you take a piece of it that is single valued? Sure. Just excise a maximum part that doesn't overlap. That part then becomes the imaginary component of a branch of log(z) and the jump. the gap, between the edges is the branch cut and the origin is the singular point called the branch point.

    Hi, Im new here and like math. :)
     
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