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Conformal Mappings

  1. Mar 1, 2014 #1
    1. The problem statement, all variables and given/known data
    Hi

    Is there a rigorous way to find conformal mappins? Say I would like to find how [tex]\phi(z)=z^{0.5}[/tex] maps the domain [itex]r\exp(i\phi)[/itex] (with [itex]r>0[/itex] and [itex]0\leq \phi \leq \pi[/itex]), how would I do this?

    Thanks in advance.
     
  2. jcsd
  3. Mar 1, 2014 #2

    pasmith

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    Homework Helper

    Please don't use the same symbol for two different objects in the same context! Either [itex]\phi[/itex] is a complex function or it's the argument of a complex number. Choose one and stick with it, and find a different symbol for the other.

    To answer your question: Start with [itex]\phi(re^{i\theta}) = r^{1/2}e^{i\theta/2}[/itex]. What values can [itex]r^{1/2}[/itex] take if [itex]r > 0[/itex]? What values can [itex]\theta/2[/itex] take if [itex]0 \leq \theta \leq \pi[/itex]? What region of the complex plane does that give you?
     
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