# Finding Branch Points & Cuts for 3 Functions

• Matterwave
In summary: Your Name]In summary, to find branch points for a complex function, we need to look at the behavior of the function as we approach different points in the complex plane. Branch points are points where the function becomes multi-valued. To introduce a single-valued branch of the function, we can choose a line or curve that connects the branch points. For function a), the branch points are z = 3 and z = -3, and for function c), the branch point is z = 1. For function b), the square root function itself is already multi-valued, so no branch cut is needed.
Matterwave
Gold Member

## Homework Statement

Find the branch points and introduce branch cuts for the below functions:

a) $$\frac{1}{4+\sqrt{z^2-9}}$$

b) $$\sqrt{4+\sqrt{z^2-9}$$

c) $$ln[5+\sqrt{\frac{z+1}{z-1}}]$$

## The Attempt at a Solution

So, the professor explained branches and branch cuts like they were completely obvious to him (which, they probably are...to him). He didn't show us basically at all how to find branching points and branch cuts except by giving us an example and finding it "obviously". Like he said the function f(z)=sqrt(z) branches at 0...but he never told us how we should find that point. I have no idea how to show where the branch points are...are they points where the function evaluates to 0 or something? Someone help?

Dear student,

Branch points and branch cuts are important concepts in complex analysis that can be a little tricky to understand at first. Essentially, a branch point is a point in the complex plane where a function becomes multi-valued, and a branch cut is a line or curve in the complex plane that is used to define a single-valued branch of the function.

To find branch points for the functions given in the forum post, we need to look at the behavior of the functions as we approach different points in the complex plane. For function a), the denominator 4 + sqrt(z^2 - 9) will be equal to 0 when z = 3 or z = -3. These points are branch points because as we approach them from different directions, the value of the function will change. To introduce a branch cut, we can choose a line or curve that connects the two branch points, such as the real axis between -3 and 3. This will define a single-valued branch of the function.

For function b), the square root function will have branch points at z = 3 and z = -3, just like in function a). However, since the square root function itself is already multi-valued, we do not need to introduce a branch cut.

For function c), we need to look at the behavior of the function as z approaches infinity. As z gets very large, the term (z+1)/(z-1) will approach 1, and the square root will be defined. Therefore, the function will be single-valued at z = infinity. However, as z approaches 1 from different directions, the value of the function will change, so we have a branch point at z = 1. To introduce a branch cut, we can choose a line or curve that connects z = 1 to z = infinity, such as the positive real axis. This will define a single-valued branch of the function.

I hope this helps to clarify the concept of branch points and branch cuts. If you have any further questions, please don't hesitate to ask. Good luck with your studies!

## 1. What is the purpose of finding branch points and cuts for 3 functions?

The purpose of finding branch points and cuts for 3 functions is to identify the points on a complex function where the function becomes multivalued or discontinuous. This information is important in understanding the behavior of the function and can help in solving complex equations.

## 2. How do you find branch points and cuts for a function?

To find branch points and cuts for a function, you need to first factorize the function and look for points where the function becomes multivalued or has a non-removable discontinuity. These points can be found by setting the denominator of the function equal to zero and solving for the variable. The solutions obtained are the branch points and the points where the function will have a cut.

## 3. Why is it important to find all branch points and cuts for a function?

It is important to find all branch points and cuts for a function because these points greatly affect the behavior of the function. They can lead to singularities, where the function is undefined, and can also affect the convergence of series expansions. By identifying these points, we can better understand the function and make accurate predictions about its behavior.

## 4. Can a function have an infinite number of branch points?

Yes, a function can have an infinite number of branch points. This is particularly true for complex functions, where the behavior can become very complex and unpredictable. It is important to identify as many branch points as possible to fully understand the function.

## 5. How do branch points and cuts affect the analyticity of a function?

Branch points and cuts can greatly affect the analyticity of a function. If a function has a branch point, it means that it is not analytic at that point and cannot be represented by a Taylor series expansion. Similarly, if a function has a cut, it will not be analytic along that line. This means that the function will not be differentiable at these points, which can greatly impact the overall analyticity of the function.

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