Results 1 to 2 of 2

Thread: Holomorphic functions with constant modulus

  1. #1
    Junior Member
    Joined
    Mar 2009
    Posts
    38

    Holomorphic functions with constant modulus

    Problem Statement: Suppose that $\displaystyle f$ is holomorphic on a domain $\displaystyle \Omega \subseteq \mathbb{C}$ with $\displaystyle |f|$ constant. Prove that $\displaystyle f$ is a constant map.

    Ideas: If $\displaystyle |f| = 0$ then $\displaystyle f = 0$. Suppose $\displaystyle |f| = b \in \mathbb{C}$, $\displaystyle b \ne 0$. So $\displaystyle f(z) = b e^{i \theta(z)}$ where $\displaystyle \theta(z) = arg_0(f(z))$ with $\displaystyle arg_0(f(z))$ is a holomorphic branch of $\displaystyle arg(f(z))$; if such a branch did not exist then $\displaystyle f$ would not be holomorphic on $\displaystyle \Omega$. I tried applying the definition of the complex derivative directly to show that $\displaystyle f' = 0$ to no avail. If it were shown that for any $\displaystyle z_1 \ne z_2$ in $\displaystyle \Omega$ that $\displaystyle \theta(z_1) = \theta(z_2)$ it would be done, but I don't see how to make that work either.

    Any hints, ideas, or a solution are welcome at this point. Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    If you want a one-liner use the open mapping theorem, otherwise use the C-R equations on the identity $\displaystyle u^2+v^2=|f|^2=c$ where $\displaystyle f(z)=u(z)+iv(z)$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Why are Holomorphic functions so important?
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: Mar 23rd 2010, 11:57 AM
  2. [SOLVED] quotient rule on holomorphic functions
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Feb 9th 2010, 11:45 AM
  3. Complex differentiable and holomorphic functions
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Dec 7th 2009, 04:03 PM
  4. Holomorphic functions
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Dec 7th 2009, 07:01 AM
  5. Implicit function theorem for holomorphic functions
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Jul 20th 2009, 01:49 PM

Search Tags


/mathhelpforum @mathhelpforum