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Help: want to find an analytical soln to this =p

  1. Nov 7, 2007 #1
    So, here's the motivation to my problem:

    I have this following equation:

    [tex]P(u) = \frac{u^n}{1+u+u^2+...+u^n}[/tex]

    , 'n' being a given constant.

    I figured out that I could simplify the power series on the denominator, and get a better solution for P(u):

    [tex]P(u) = \frac{u^n-u^{n+1}}{1 - u^{n+1}}[/tex]


    Now, I am interested in P(u) = x, and 'x' is a number between 0 and 1.
    What I have been struggling with is finding an analytical solution for u(x).
    To make it more clear, I would like to know what 'u' will equal, for a given 'x' value, and a known 'n'.

    I've tried a lot of different approaches, but I just can't seem to come up with an analytical solution for u(x).

    One attempt and where I get stuck: [tex]u^n + u^{n+1}(x-1) = x[/tex]


    Any help would be appreciated. I would like to be enlightened. =p

    Thanks,

    Mark
     
    Last edited: Nov 7, 2007
  2. jcsd
  3. Nov 7, 2007 #2

    Gib Z

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    Homework Helper

    I am inclined to say there is no analytical solution for it, though I have no proof, just instinct.
     
  4. Nov 7, 2007 #3
    You can use the inverse function theorem along with http://planetmath.org/encyclopedia/DerivativeOfInverseFunction.html [Broken] to find the derivative of u(x) from what you have (x(u)).
     
    Last edited by a moderator: May 3, 2017
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