Branches of the complex logarithm

In summary, the student attempted to find a branch of log(z) that is analytic inside the unit circle, but ran into trouble because the function is not analytic at the angle \tau on the unit circle.
  • #1
JonathanT
18
0

Homework Statement


Find a branch of log(z − 1) that is analytic inside the unit circle. What is the value of this branch at z = 0?

2. The attempt at a solution

Alright so clearly the log(z) function is analytic at all points accept for the negative real axis.

So log(z-1) will be analytic at all points x ≤ 1. My problem is choosing a branch. I'm really bad at doing this. I understand if the function was something like log(z+1) it would already be analytic inside the unit circle using the principle branch. I just don't see how I can choose a branch that is analytic inside the unit circle. I think if I understand this problem I can be prepared for my test tomorrow. Thanks for any help in advance.
 
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  • #2
I just don't see how I can choose a branch that is analytic inside the unit circle.
As just the real axis is problematic, you can begin in the upper [or lower] part of the plane, choose a branch there, and try to extend this to the interior of the unit circle afterwards.
 
  • #3
So if I understand right from what I've got in my notes I can just start my branch at some arbitrary point say something like.

[itex]\tau[/itex]<arg(w)≤[itex]\tau[/itex]+2[itex]\pi[/itex]

Where [itex]\tau[/itex] = [itex]\pi[/itex]/4 maybe?

However, then the function would not be analytic at the angle [itex]\tau[/itex] on the unit circle am I right?
 
  • #4
Tell me if this works:

log(z-1) = log(-1) + log(1-z) = Log|-1| + i[itex]\cdot[/itex]arg(-1) + log(1-z)

= 0 + i[itex]\cdot[/itex][itex]\pi[/itex] + log(1-z)

So I end up with f(z) = log(1-z) + i[itex]\cdot[/itex][itex]\pi[/itex]

Then I could use the principle argument because now the cut is at real values of x ≥ 1. Which is outside the inside of the unit circle. I have no idea if my logic here is right though.
 
  • #6
Awesome! I think I finally get this. Hope I do well on the test today. Thanks!
 

FAQ: Branches of the complex logarithm

What are the branches of the complex logarithm?

The branches of the complex logarithm refer to the different values of the logarithm function in the complex plane. Unlike the real logarithm function, which has a single value for each input, the complex logarithm has an infinite number of possible values.

How is the complex logarithm different from the real logarithm?

The complex logarithm is different from the real logarithm in that it takes complex numbers as inputs and has an infinite number of possible values. In contrast, the real logarithm only takes positive real numbers as inputs and has a single value for each input.

What is the principal branch of the complex logarithm?

The principal branch of the complex logarithm is the branch that is commonly used and defined as the natural logarithm of the complex number with the smallest positive argument. It is denoted as Log(z) and has a range of (-π, π].

How are the branches of the complex logarithm related?

The branches of the complex logarithm are related through the concept of branch cuts. A branch cut is a line or curve in the complex plane where the function is not defined, and it separates the different branches of the complex logarithm. The branches are also related through a periodicity property, where each branch is a rotation of the other branches.

How are the branches of the complex logarithm used in applications?

The branches of the complex logarithm are used in various applications in mathematics, physics, and engineering. Some examples include solving complex equations, calculating the complex roots of a number, and analyzing the behavior of complex systems. They are also used in the study of complex analysis and the theory of Riemann surfaces.

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