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Help me solve this equation involving exponentials!

  1. Mar 4, 2010 #1
    I'm trying to solve the following equation for [tex]z\in \mathbb C \setminus \{ 0 \}[/tex], [tex]w\in \mathbb C[/tex]:

    e^{1/z} + \frac{1}{1-e^{1/z}} = w.

    How in the world should I go about doing that?
  2. jcsd
  3. Mar 4, 2010 #2


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    Let u= [itex]e^{1/z}[/itex]. Then your equation becomes
    [tex]u+ \frac{1}{1- u}= w[/tex]
    Multiply both sides by 1- u to get u(1- u)+ 1= w(1- u) or [itex]u- u^2+ 1= w- uw[/itex] which equivalent to the quadratic equation [itex]u^2- (1+w)u+ w-1= 0[/itex]. Use the quadratic formula to solve that, then solve [itex]e^{1/z}= u[/math] by taking the logarithm of both sides.
  4. Mar 4, 2010 #3
    Great. Thanks. This was my original approach, but for some reason I wasn't sure if I could make that change of variables and apply the quadratic formula like you did. But you've confirmed my intuition, so I'm going with it!
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