Find a solution to $z+e^{-z}=a$ where $a>1$.

Question

Find a solution to $z+e^{-z}=a$ where $a>1$.

I have tried many manipulations with little success. I don't see how I can solve this for $z$. Any solutions or hints are greatly appreciated. I think that there is one real solution to this equation and that it will be in the right half plane.

Answer 1

There is no elementary solution to that. Wolfram Alpha produces $$z = W(-e^{-a})+a $$ which involves the Lambert W function, but this probably doesn't make you much wiser in practice -- you would just solve it numerically.

Answer 2

As noted in the comments, you can write $$z=W(-e^{-a})+a$$ but this fails to shed any real light on the problem. There aren't really any algebraic manipulations that you can do to work out anything useful. You can use numerical approximation techniques if you want a decimal answer though.

Answer 3

This is what's known as a transcendental equation, which has no elementary closed form. It can, however be solved in terms of the Lambert W function.

$\begin{align*}z + e^{-z} &= a\\ze^{z} + 1 &= ae^z\\e^z(z-a) &= -1\\e^{z-a}(z-a)&= -e^{-a}\\ z-a &= W(-e^{-a})\\z &= W(-e^{-a}) + a\end{align*}$