1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex function question

  1. Sep 12, 2006 #1
    Problem: Suppose [tex] \Omega \in \mathbb{C} [/tex] is open and connected, [tex] f [/tex] is differentiable on [tex] \Omega [/tex], and [tex] f(z) \in \mathbb{R} , \ \forall z \in \Omega [/tex]. Prove that [tex] f(z) [/tex] is constant.

    Is this just a matter of solving the Cauchy-Riemann equations? If so, I think the proof is relatively straightforward.

    Since the function is differentiable, it can be written as [tex] f(z) = u(x,y) + iv(x,y) [/tex] where u and v are differentiable real-valued functions.

    Since [tex] f(z) [/tex] is real, [tex] v(x,y) = 0 [/tex].

    Then, the first CR equation tells us that [tex] \frac{ \delta u}{ \delta x} = \frac{ \delta v}{ \delta y} [/tex]. But, since [tex] v(x,y) = 0 [/tex], both of these fractions equal 0.

    Next, the second CR equation tells us that [tex] \frac{ \delta u}{ \delta y} = - \frac{ \delta v}{ \delta x} [/tex]. But, since [tex] v(x,y) = 0 [/tex], both of these fractions equal 0.

    So, if [tex] \frac{ \delta u}{ \delta x} = 0 = \frac{ \delta u}{ \delta y} [/tex], then [tex] u(x,y) [/tex] must be some constant function.

    Is that all I need?
  2. jcsd
  3. Sep 12, 2006 #2
    It looks ok yes.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Complex function question
  1. Complex function (Replies: 17)