Cfrgtkky

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