Math Help - Complex Analysis-Analytic Functions

1. Complex Analysis-Analytic Functions

1. Prove that there is no analytic function inside
Code:
z:0<|z|<1
such as every z in this region satysfies:
Code:
f^2(z)=z
.

2. Let f be analytic at the annulus:
Code:
 D=z:0<|z-a|<r
.
Prove that if f has an antideriative function then Res(f,a)=0 .

Hope you'll be able to help me

Thanks a lot!

2. Regarding #2, what is a "former" function?

3. iI meant antideriative...sry

4. For #1, suppose that such a function exists, and show that the Cauchy-Riemann equations are not satisfied at z=0.
Or you can differentiate implicitly with respect to $z$ and show that $f'(z)$ cannot be analytic at $0$.

5. For #2, suppose that an antiderivative exists. Show that this implies that the integral along a simple closed contour inside the annulus vanishes... so what if the interior of the contour contains the point $z=0$?

We can't use Cauchy-Riemann,because f isn't analytic at z=0...
How can we solve this problem?

2 is completely understandable...

Thanks @!

7. Originally Posted by WannaBe
We can't use Cauchy-Riemann,because f isn't analytic at z=0...
How can we solve this problem?

2 is completely understandable...

Thanks @!
I'm sorry, I hadn't seen the point 0 was excluded. I hadn't had my coffee just yet!

One way which I can think of is this : a function analytic inside a domain maps a closed contour to a closed contour. So assume there exists such a function. Now take a simple closed contour around the origin, and show that $f$ does not map it to a closed contour (use the argument principle, and the fact that $2f(z)f'(z)=1$ inside the punctured disc).

8. Thanks...