Complex Analysis: Analytic Function F(z)?

In summary, the conversation is about checking if there is an analytic function F(z) in a given area with a specific derivative. The participant suggests that if the derivative has an antiderivative in the set, its integral around any closed curve will be zero. However, the other participant points out that the antiderivative will involve Ln, which is not analytic in the set. Further clarification is requested.
  • #1
asi123
258
0

Homework Statement



Hey guys.
I have this question, I took it from a test.
I need to check if there is an analytic function F(z) in this area (in the pic) that has this derivative (in the pic).

http://img256.imageshack.us/img256/7826/25453238.jpg

Well, the derivative is analytic in this area, right?
If the derivative is analytic, shouldn't F(z) be analytic and even more important, can it be that easy ?
I'm not sure about the answer, I could really use some help.

Thanks a lot.


Homework Equations





The Attempt at a Solution

 
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  • #2
If F'(z) has an antiderivative in the set described, then its integral around any closed curve in that set will be zero. Are you sure that happens? Consider what form an antiderivative of F'(z) would have, and what functions would be involved.
 
  • #3
Wretchosoft said:
If F'(z) has an antiderivative in the set described, then its integral around any closed curve in that set will be zero. Are you sure that happens? Consider what form an antiderivative of F'(z) would have, and what functions would be involved.

The antiderivative of F'(z) will probably involve Ln, and Ln is not analytic in this set.
Is that what you mean?

Thanks a lot.
 

1. What is a complex analytic function?

A complex analytic function, also known as a holomorphic function, is a complex-valued function of a complex variable that is differentiable at every point in its domain. This means that the function has a well-defined derivative at every point, and can be expanded into a power series.

2. What is the importance of analytic functions in complex analysis?

Analytic functions are important in complex analysis because they are the most well-behaved functions in the complex plane. They have many useful properties, such as being infinitely differentiable and having unique analytic continuations. Analytic functions also play a crucial role in applications such as physics and engineering.

3. How do you determine if a function is analytic?

To determine if a function is analytic, you can use the Cauchy-Riemann equations, which state that for a function f(z) = u(x,y) + iv(x,y), where z = x + iy, the partial derivatives of u and v must satisfy the conditions ux = vy and uy = -vx. If these conditions are met, then the function is analytic.

4. What is the relationship between complex analytic functions and conformal mappings?

In complex analysis, conformal mappings are functions that preserve angles between intersecting curves. It can be shown that any analytic function that is non-constant and conformal is also a conformal mapping. This means that complex analytic functions play an important role in the study of conformal mappings.

5. Can a function be analytic at some points and not others?

Yes, a function can be analytic at some points and not at others. This is because the Cauchy-Riemann equations only provide a necessary condition for a function to be analytic, not a sufficient one. A common example is the function f(z) = |z|2, which is analytic everywhere except at the origin.

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