Can someone find this inverse (and solve Burger's equation)?











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My question regards the following function:



$f(x) = begin{cases}frac{x}{logleft(frac{x+1}{x-1}right)}&text{ for }x>1,\
0&text{ for x=1}end{cases}$



which arose in circumstances I will describe below. Quite simply, I'd like to know if someone could find an expression for $f^{-1}$ of any kind, be it in closed form or not (e.g., a power series or an integral).



My motivation for this question has got to do with the steady-state viscous Burger's equation



$kappaddot{w}(x) = w(x)dot{w}(x)$ with $xin[-1,1]$ and $w(-1)=-w(1) = 1$ and $kappa>0$.



After some deliberation I gathered that a solution is given by



$w(x) = ccdot frac{e^{-frac{c}{kappa} y} - 1}{e^{-frac{c}{kappa} y}+1}$



where $c$ is chosen so that the boundary conditions hold, which is equivalent to, lo and behold,



$kappa = frac{c}{logleft(frac{c+1}{c-1}right)}$.



Also, references to relevant literature are appreciated. I have mostly relied on Wolfram's articles on the (inverse) hyperbolic tangent:



http://mathworld.wolfram.com/HyperbolicTangent.html



and



http://mathworld.wolfram.com/InverseHyperbolicTangent.html










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    up vote
    0
    down vote

    favorite












    My question regards the following function:



    $f(x) = begin{cases}frac{x}{logleft(frac{x+1}{x-1}right)}&text{ for }x>1,\
    0&text{ for x=1}end{cases}$



    which arose in circumstances I will describe below. Quite simply, I'd like to know if someone could find an expression for $f^{-1}$ of any kind, be it in closed form or not (e.g., a power series or an integral).



    My motivation for this question has got to do with the steady-state viscous Burger's equation



    $kappaddot{w}(x) = w(x)dot{w}(x)$ with $xin[-1,1]$ and $w(-1)=-w(1) = 1$ and $kappa>0$.



    After some deliberation I gathered that a solution is given by



    $w(x) = ccdot frac{e^{-frac{c}{kappa} y} - 1}{e^{-frac{c}{kappa} y}+1}$



    where $c$ is chosen so that the boundary conditions hold, which is equivalent to, lo and behold,



    $kappa = frac{c}{logleft(frac{c+1}{c-1}right)}$.



    Also, references to relevant literature are appreciated. I have mostly relied on Wolfram's articles on the (inverse) hyperbolic tangent:



    http://mathworld.wolfram.com/HyperbolicTangent.html



    and



    http://mathworld.wolfram.com/InverseHyperbolicTangent.html










    share|cite|improve this question


























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      My question regards the following function:



      $f(x) = begin{cases}frac{x}{logleft(frac{x+1}{x-1}right)}&text{ for }x>1,\
      0&text{ for x=1}end{cases}$



      which arose in circumstances I will describe below. Quite simply, I'd like to know if someone could find an expression for $f^{-1}$ of any kind, be it in closed form or not (e.g., a power series or an integral).



      My motivation for this question has got to do with the steady-state viscous Burger's equation



      $kappaddot{w}(x) = w(x)dot{w}(x)$ with $xin[-1,1]$ and $w(-1)=-w(1) = 1$ and $kappa>0$.



      After some deliberation I gathered that a solution is given by



      $w(x) = ccdot frac{e^{-frac{c}{kappa} y} - 1}{e^{-frac{c}{kappa} y}+1}$



      where $c$ is chosen so that the boundary conditions hold, which is equivalent to, lo and behold,



      $kappa = frac{c}{logleft(frac{c+1}{c-1}right)}$.



      Also, references to relevant literature are appreciated. I have mostly relied on Wolfram's articles on the (inverse) hyperbolic tangent:



      http://mathworld.wolfram.com/HyperbolicTangent.html



      and



      http://mathworld.wolfram.com/InverseHyperbolicTangent.html










      share|cite|improve this question















      My question regards the following function:



      $f(x) = begin{cases}frac{x}{logleft(frac{x+1}{x-1}right)}&text{ for }x>1,\
      0&text{ for x=1}end{cases}$



      which arose in circumstances I will describe below. Quite simply, I'd like to know if someone could find an expression for $f^{-1}$ of any kind, be it in closed form or not (e.g., a power series or an integral).



      My motivation for this question has got to do with the steady-state viscous Burger's equation



      $kappaddot{w}(x) = w(x)dot{w}(x)$ with $xin[-1,1]$ and $w(-1)=-w(1) = 1$ and $kappa>0$.



      After some deliberation I gathered that a solution is given by



      $w(x) = ccdot frac{e^{-frac{c}{kappa} y} - 1}{e^{-frac{c}{kappa} y}+1}$



      where $c$ is chosen so that the boundary conditions hold, which is equivalent to, lo and behold,



      $kappa = frac{c}{logleft(frac{c+1}{c-1}right)}$.



      Also, references to relevant literature are appreciated. I have mostly relied on Wolfram's articles on the (inverse) hyperbolic tangent:



      http://mathworld.wolfram.com/HyperbolicTangent.html



      and



      http://mathworld.wolfram.com/InverseHyperbolicTangent.html







      real-analysis differential-equations






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      edited Nov 15 at 23:02

























      asked Nov 15 at 16:42









      IAnemaet

      385




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