# Math Help - Complex analysis

1. ## Complex analysis

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.

2. The integral of $\frac z{z^2+4}$ is $\tfrac12\ln(z^2+4)$, and this function cannot be defined globally on the set $\mathbb{C}\setminus\{\pm2i\}$ (logarithm functions can usually only be defined on some kind of cut plane).