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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.

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    $\begingroup$ en.wikipedia.org/wiki/Lambert_W_function $\endgroup$ – Dark Malthorp May 24 '18 at 20:20
  • $\begingroup$ There should be plenty to go on there, and in the references. $\endgroup$ – Dark Malthorp May 24 '18 at 20:20
  • $\begingroup$ The root of an expression is a value of the variable that makes the expression zero. Expressions have roots, equations have solutions. $\endgroup$ – Acccumulation May 24 '18 at 20:28
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You might write your equation as $$ a = - z e^z $$

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

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