Comlplex analysis problem

1. May 15, 2007

sparkster

So let f be analytic in the open unit disk and continuous on the closed unit disk. Also, |f(z)|=1 for |z|=1, all zeros are simple zeros at 0, and f'(0)=-1/2.

I need to find f.

I've tried using the cauchy integral formula for f' but that's not getting me anywhere. Can anyone point me in the right direction?

2. May 15, 2007

ObsessiveMathsFreak

Perhaps you could use the fact that the real an imaginary parts of a complex differentiable function are harmonic, i.e.
$$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} = \frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2}= 0$$
in the disc.

Also on the boundary
$$|f|=1=|f|^2=u^2+v^2=1$$

And for the derivative you have
$$f'(0)=\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}=\frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y}=-\frac{1}{2}$$

This gives
$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}=-\frac{1}{2}$$
$$\frac{\partial v}{\partial x}=\frac{\partial u}{\partial y}=0$$
At the origin.

But is this enough to solve the partial differential equation?