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How to solve for x in this problem

  1. Jul 29, 2010 #1
    what's the general solution for x in the following equation

    [tex] x=c-N^{-\left(\frac{1+x}{1-x}\right)}[/tex]

    i could solve using numerical methods , but i need an exact-explicit solution .
     
  2. jcsd
  3. Jul 30, 2010 #2

    CRGreathouse

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    Looks like you'll need Lambert's W, then.
     
  4. Jul 30, 2010 #3
    You could substitute the brakets with x by a new variable and then try to rearrange until you can apply
    http://mathworld.wolfram.com/LambertW-Function.html
    In the end this new function is just like numeric. But at least you have a name for it ;)
     
  5. Jul 30, 2010 #4
    thanks for the reply .. i tried this :
    substitute

    [tex]y=\frac{1+x}{1-x}[/tex]


    then

    [tex]N^{-y}=c-\frac{y-1}{y+1}[/tex]

    or

    [tex](y+1)N^{-y}=(c-1)y+(c+1)[/tex]

    but then i got a mental block ... i don't know how to transform the problem into the form :

    [tex]A(N,c)=B(x) e^{B(x)} [/tex]
     
  6. Aug 1, 2010 #5
    i managed to reduce the problem down to the form :

    [tex]z^{(z-A)}=B[/tex]

    again , i'm stuck !!
     
  7. Aug 1, 2010 #6
    Hmm, maybe it's not doable with that function alone. I only get
    [tex]
    W(x)=Ax+B
    [/tex]
    But anyway, what's the difference between numerics and knowing an expression with W? W has to be computed numerically anyway (just like sin, cos and all other functions).
     
  8. Aug 1, 2010 #7

    Mentallic

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    [tex]\pi[/tex] has to be computed numerically too but we still allow it a symbol for convenience. And I would much rather sin(2) over a numerical approximation for it, even though if anyone wanted to evaluate sin(2) they would end up with an approximation anyway.

    I also couldn't find a solution in terms of the W function. I checked with Wolfram and it too couldn't find one.
     
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