Branch cuts

  • Thread starter hoffmann
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  • #1
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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.
 

Answers and Replies

  • #2
olgranpappy
Homework Helper
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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.
perhaps the point at infinity?
 
  • #3
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^^ what do you mean by this?
 
  • #4
olgranpappy
Homework Helper
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^^ what do you mean by this?
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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