How to solve for x in this problem

[mmzaj]

what's the general solution for x in the following equation

$$x=c-N^{-\left(\frac{1+x}{1-x}\right)}$$

i could solve using numerical methods , but i need an exact-explicit solution .

 

[CRGreathouse]

Looks like you'll need Lambert's W, then.

 

[Gerenuk]

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 ;)

 

[mmzaj]

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 ;)

thanks for the reply .. i tried this :
substitute

$$y=\frac{1+x}{1-x}$$then

$$N^{-y}=c-\frac{y-1}{y+1}$$

or

$$(y+1)N^{-y}=(c-1)y+(c+1)$$

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

$$A(N,c)=B(x) e^{B(x)}$$

 

[mmzaj]

i managed to reduce the problem down to the form :

$$z^{(z-A)}=B$$

again , I'm stuck !

 

[Gerenuk]

Hmm, maybe it's not doable with that function alone. I only get
$$W(x)=Ax+B$$
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).

 

[Mentallic]

$$\pi$$ 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.