Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Comlplex analysis problem

  1. May 15, 2007 #1
    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. jcsd
  3. May 15, 2007 #2
    Perhaps you could use the fact that the real an imaginary parts of a complex differentiable function are harmonic, i.e.
    [tex]\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[/tex]
    in the disc.

    Also on the boundary

    And for the derivative you have
    [tex]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}[/tex]

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

    But is this enough to solve the partial differential equation?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Comlplex analysis problem
  1. Analysis problem (Replies: 3)

  2. Analysis problem (Replies: 2)