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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
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