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Equation with complex variable

  1. Apr 12, 2010 #1
    1. The problem statement, all variables and given/known data

    Do you know how to find the solution of the equation:

    a - z - exp(-z) = 0 , where a > 1 and z is a complex variable

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 12, 2010 #2
    [tex] Xe^X=Y \iff W(Y)=X [/tex]

    For example

    [tex]e^{-z}=a-z[/tex]

    try to show that

    [tex]C=f(z)e^{f(z)}[/tex]

    where C is a constant and you will get the right answer

    W(y) is Lambert W function
     
  4. Apr 13, 2010 #3
    I couldn't write it in the form :

    C = f(x) exp (fx))

    Also I don't see how this will give me the right answer.
     
  5. Apr 13, 2010 #4

    lanedance

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    Homework Helper

    how about this
    [tex]e^{-z}=a-z[/tex]
    [tex]-1= e^{z}(z-a)[/tex]
    [tex]-e^{-a}= e^{z-a}(z-a)[/tex]
     
  6. Apr 13, 2010 #5

    lanedance

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    Homework Helper

    Last edited: Apr 13, 2010
  7. Apr 13, 2010 #6
    So the solution is:

    -e^(-a)=(z-a) e^(z-a)
    W(-e^(-a) )=z-a
    z=W(-e^(-a) )+a=-a+a=0

    am I right or not?

    If so, the answer according ot the question should be in the have plane Re z >= 0

    and must be real.

    What happen to the solution if a goes to 1.

    from the statement of the question I can guess that my answer is not on the right way
     
  8. Apr 14, 2010 #7

    HallsofIvy

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    Staff Emeritus
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    Then what was the statement of the question?
     
  9. Apr 14, 2010 #8
    The full statement:

    Let a >= 1 then show that the given equation has exactly one solution in the half plane Rez>= 0, and that solution is real. What happen to the solution if a goes to 1?
     
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