Help me solve this equation involving exponentials

[AxiomOfChoice]

I'm trying to solve the following equation for z\in \mathbb C \setminus \{ 0 \}, w\in \mathbb C:

<br /> e^{1/z} + \frac{1}{1-e^{1/z}} = w.<br />

How in the world should I go about doing that?

 

[HallsofIvy]

Let u= e^{1/z}. Then your equation becomes
u+ \frac{1}{1- u}= w
Multiply both sides by 1- u to get u(1- u)+ 1= w(1- u) or u- u^2+ 1= w- uw which equivalent to the quadratic equation u^2- (1+w)u+ w-1= 0. Use the quadratic formula to solve that, then solve e^{1/z}= u[/math] by taking the logarithm of both sides.

 

[AxiomOfChoice]

Let u= e^{1/z}. Then your equation becomes
u+ \frac{1}{1- u}= w
Multiply both sides by 1- u to get u(1- u)+ 1= w(1- u) or u- u^2+ 1= w- uw which equivalent to the quadratic equation u^2- (1+w)u+ w-1= 0. Use the quadratic formula to solve that, then solve e^{1/z}= u[/math] by taking the logarithm of both sides.

<br /> <br /> Great. Thanks. This was my original approach, but for some reason I wasn&#039;t sure if I could make that change of variables and apply the quadratic formula like you did. But you&#039;ve confirmed my intuition, so I&#039;m going with it!