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Complex functions

  1. Jul 23, 2007 #1
    1. The problem statement, all variables and given/known data
    Find the complex function of z^(1/2))=(x+iy)^(1/2)


    3. The attempt at a solution
    The first step is z^(1/2)=e^((1/2)ln(z))=e^(1/2)[(ln|z|+i(theta)+2((pi)n)]

    But the answers were not in this form.
     
  2. jcsd
  3. Jul 23, 2007 #2

    Gib Z

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    What is your question asking? What do you mean by, find the complex function...
     
  4. Jul 23, 2007 #3

    CompuChip

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    Do you want to extend the square root to the complex plane?
    And what form were the answers in? Perhaps they are the same (just written down differently)?
     
    Last edited: Jul 23, 2007
  5. Jul 23, 2007 #4
    The definition of [tex]a^b[/tex] for [tex]a,b\in \mathbb{C}[/tex] and [tex]a\not =0[/tex] is defined as [tex]\exp (b\ln a)[/tex].

    So, [tex]z^{1/2} = \exp \left( \frac{\log z}{2} \right) = \exp \left( \frac{\ln |z|}{2} + i\cdot \frac{\arg z}{2} \right) = \sqrt{|z|}\cdot e^{i\arg(z)/2}[/tex]
     
  6. Jul 24, 2007 #5
    Sorry, just to clarify the question is asking to find u(x,y) and v(x,y) where z^(1/2)=u(x,y)+iv(x,y) where z=x+iy.
     
  7. Jul 24, 2007 #6

    CompuChip

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    That's called the real and imaginary part.
    Recall Euler's identity
    [tex]e^{i \phi} = \cos \phi + i \sin\phi.[/tex]
    Will that do?
     
  8. Jul 24, 2007 #7
    [tex]\sqrt{|z|}\cos \left( \frac{\arg z}{2} \right) + i \sqrt{|z|}\sin \left( \frac{\arg z}{2} \right)[/tex]
     
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