Understanding Branch Cuts for Analytic Functions in the Complex Plane

In summary, the function f(z) = log(z2 - 1) in the complex plane cannot be defined with a simple branch cut connecting z = -1 and z = +1 due to the point at infinity. However, the function f(z) = (z^2 - 1)^(1/2) can be defined with this branch cut, as shown by analyzing the behavior of the function as it goes around the branch cut.
  • #1
hoffmann
70
0
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.
 
Physics news on Phys.org
  • #2
hoffmann said:
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
^^ what do you mean by this?
 
  • #4
hoffmann said:
^^ 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.
 

Related to Understanding Branch Cuts for Analytic Functions in the Complex Plane

1. What is a branch cut in the context of analytic functions in the complex plane?

A branch cut is a path or curve in the complex plane along which a function is not defined or is discontinuous. It is typically chosen to be the shortest path between two branch points, where the function takes on different values. Branch cuts are used to define branches of multivalued functions, such as the complex logarithm or square root, in order to make them single-valued and well-defined.

2. How do branch cuts affect the behavior of analytic functions in the complex plane?

Branch cuts can cause discontinuities in the behavior of analytic functions, leading to different values for the function depending on which branch of the function is chosen. They also affect the domain of the function, as points on the branch cut are not included in the domain.

3. How are branch cuts related to branch points?

Branch cuts are typically chosen to connect two branch points, which are points in the complex plane where the function takes on different values. The choice of branch cut determines the branches of the function and the behavior of the function near the branch points.

4. Can you give an example of a function with a branch cut?

The complex logarithm function is a commonly used example of a function with a branch cut. The branch cut is typically chosen to be the negative real axis, connecting the branch points at 0 and infinity. In this case, the function is discontinuous along the negative real axis, and the values of the logarithm function differ on either side of the branch cut.

5. How can I determine the location and behavior of branch cuts for a given analytic function?

The location and behavior of branch cuts can be determined by examining the singularities of the function, such as branch points and poles. The choice of branch cut is not unique and can vary depending on the specific function and the desired properties of the branches. In some cases, a branch cut may need to be chosen to avoid other singularities or to ensure the function is well-defined and single-valued.

Similar threads

Replies
3
Views
2K
Replies
3
Views
628
Replies
2
Views
577
Replies
3
Views
3K
Replies
13
Views
2K
Replies
14
Views
959
Back
Top