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.

here a more general case. – Mhenni Benghorbal Jul 07 '14 at 18:10 — Mhenni Benghorbal, Jul 07 '14 at 18:10

score 6 · Accepted Answer · answered Jul 07 '14 at 17:30

6

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.

answered Jul 07 '14 at 17:30

Antonio Vargas

Thanks, wikipedia said :$W\left(-1\right) \approx -0.31813-1.33723{\rm{i}}$, did you know how can we prove this ? – Jul 07 '14 at 17:34
Use Newton's method on the equation $z=e^z$ with an initial guess like $0.3+1.3i$. – Antonio Vargas Jul 07 '14 at 17:36
Newton's method, did you mean that http://en.wikipedia.org/wiki/Newton%27s_method ? I am unfamiliar with it if you can expand your answer (if you have time ). – Jul 07 '14 at 17:39
1

Yes, that's the right wiki page. Newton's method is an iterative process for finding numerical approximations to solutions of equations. I definitely recommend reading about it - any calculus book should have a good description of the process. – Antonio Vargas Jul 07 '14 at 17:46
Thanks! I will read about it so. (BTW +1 even if is a community wiki) – Jul 07 '14 at 18:19

1 Answers1