Analysis of all the roots of the equation : $ z = -ae^{-z}, a > 0$.

Question

I want to analyze all roots of the equation: $ z = -ae^{-z}, a > 0 $. If somebody gives a method or suitable reference is greatly appreciated.

I found in a textbook, this has two real solutions when $a< e^{-1}$. Say $z_2, z_1$. Write $z_2 < z_1 <0$. Then it mentioned that all other roots $z$ satisfies $Re(z)< z_2$. I'm trying to prove it. Also, it doesn't have real solution when $a> e^{-1}$.

Also, I want to know, is there a way to represent solutions (especially complex solution), at least using an approximated formula?

Thank you.

Answer 1

You might write your equation as $$ a = - z e^z $$

If you're interested in real roots, consider the graph of $f(z) = - z e^z$.