Complex Analysis Fun: Analytic Antiderivatives in {z:|z|>2}

In summary, the problem asks to show that the function \frac{z}{(z-1)(z-2)(z+1)} has an analytic antiderivative in the region \{z \in \bold{C}:|z|>2\}. The Attempt at a Solution suggests using a partial fraction decomposition and logarithms to create a function that is analytic in the region, but notes that this may not be the desired outcome. Further attempts involve using specific branch cuts of log, but there is uncertainty about their validity.
  • #1
Mystic998
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Homework Statement


Show that [tex] \frac{z}{(z-1)(z-2)(z+1)}[/tex] has an analytic antiderivative in [itex]\{z \in \bold{C}:|z|>2\}[/itex]. Does the same function with z^2 replacing z (EDIT: I mean replacing the z in the numerator, not everywhere) have an analytic antiderivative in that region?


Homework Equations


Um lots of things I imagine.


The Attempt at a Solution


Well, I'm pretty sure that I can do a partial fraction decomposition in both cases, then the appropriate logarithms would give me a function that's analytic on the region minus whatever line I do the branch cut on. But unless there's some huge typo in the problem, I don't think that's what's being sought. I'm not really sure what else to do in this situation though. I have some other thoughts on the problem that may or may not work, but they're kind of long winded, and I'd rather not go into them unless I really have to. So, any suggestions?
 
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  • #2
Well, I still haven't been able to come up with anything. I thought maybe I could use specific branch cuts of log to show that the partial fraction decomposition integrates to zero around any closed curve in the region (I think it would require different branch cuts for each integral, so I don't know how valid this is), but then I have a problem if the closed curve has the complement of my region "inside" it.
 

Related to Complex Analysis Fun: Analytic Antiderivatives in {z:|z|>2}

1. What is Complex Analysis Fun: Analytic Antiderivatives in {z:|z|>2}?

Complex Analysis Fun: Analytic Antiderivatives in {z:|z|>2} is a branch of mathematics that deals with the study of functions of complex numbers and their properties. It involves the use of complex numbers, which are numbers that have both a real and imaginary component, and their behavior under operations such as addition, subtraction, multiplication, and division.

2. What are analytic antiderivatives?

Analytic antiderivatives, also known as complex antiderivatives, are functions that are the inverse of other complex functions. They represent the fundamental concepts of complex analysis and are used to solve problems involving complex numbers and their properties.

3. What is the significance of the domain {z:|z|>2} in this context?

The domain {z:|z|>2} refers to the set of all complex numbers whose absolute values are greater than 2. In Complex Analysis Fun: Analytic Antiderivatives in {z:|z|>2}, this domain is used to study the behavior of complex functions outside of a certain region, which can provide valuable insights and solutions to complex problems.

4. How are analytic antiderivatives different from regular antiderivatives?

Analytic antiderivatives are specific to complex functions and involve the use of complex numbers, while regular antiderivatives can be applied to real-valued functions. Analytic antiderivatives also have unique properties and behaviors, such as being holomorphic (differentiable everywhere in their domain) and having a power series representation.

5. What are some real-world applications of Complex Analysis Fun: Analytic Antiderivatives in {z:|z|>2}?

Complex Analysis Fun: Analytic Antiderivatives in {z:|z|>2} has many real-world applications, including in physics, engineering, and finance. It is used to model and solve complex systems, analyze electrical circuits, and predict the behavior of financial markets. It also has applications in signal processing, fluid dynamics, and quantum mechanics.

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