Branch-cut singularity of a complex logarithm

[gremezd]

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?

 

[gammamcc]

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.

 

[squidsoft]

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, I am new here and like math. :)