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.

  • 4
    $\begingroup$ 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

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

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.