1. Not finding help here? Sign up for a free 30min 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 Variables. Complex square root function

  1. Mar 1, 2008 #1
    1. The problem statement, all variables and given/known data

    Proof that [tex]f(z)=\sqrt{z}=e^{\frac{\ln z}{2}}[/tex] with logarithm branch [tex][0,2\pi)[/tex]. Then [tex]f[/tex] maps horizontal and vertical lines in [tex]A=\mathbb{C}-\{\mathbb{R}^{+}\cup\{0\}\}[/tex] on hyperbola branches.

    2. Relevant equations

    I have that [tex]\ln_{[0,2\pi)} (z)=\ln\vert z\vert+i\mathop{\rm arg}\nolimits_{[0,2\pi)} (z)[/tex]

    3. The attempt at a solution

    I have tried to proof it directly, that means to describe vertical lines with [tex]z(y)=(a,y)[/tex] and horizontal ones with [tex]w(x)=(x,a)[/tex] and substitute in [tex]f[/tex]. That produces an horrific expression that I can't reduce to an hyperbola. What can I do?
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?



Similar Discussions: Complex Variables. Complex square root function
  1. Complex analysis (Replies: 0)

Loading...