Solution of $\exp(z)=z$ in $\Bbb{C}$.

Question

I have posted a related question here. I thinkg this one is more interesting:

What about the solution of $\exp(z)=z$ in $\Bbb{C}$?

My try :

$z \mapsto e^z - z$ is entire non-constant.

Perhaps $z \mapsto e^z - z$ can be developed in Weierstrass product.

Also any numerically approach will be very interesting.

Thanks you in advance.

Answer 1

If

$$z = e^z$$

then

$$-ze^{-z} = -1$$

so

$$-z = W(-1)$$

and thus

$$z = - W(-1),$$

where $W$ is any branch of the Lambert W function.