# Help me solve this equation involving exponentials!

## Main Question or Discussion Point

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

$$e^{1/z} + \frac{1}{1-e^{1/z}} = w.$$

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 $e^{1/z}= u[/math] by taking the logarithm of both sides. Let u= [itex]e^{1/z}$. 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 [itex]e^{1/z}= u[/math] by taking the logarithm of both sides.