1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: 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


    try to show that


    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


    User Avatar
    Homework Helper

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


    User Avatar
    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


    User Avatar
    Science Advisor

    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?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook