Complex Analysis: Analytic Function F(z)?

[asi123]

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

 

[Wretchosoft]

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.

 

[asi123]

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.