e-e-x-x-x-

Question Number 128885 by naka3546 last updated on 11/Jan/21

e^{e^{x}} = x

x = ?

Commented by naka3546 last updated on 11/Jan/21

x \in R ? o r x \in C ?

Commented by mr W last updated on 11/Jan/21

e^{e^{\dots e^{x}}} = x \Leftrightarrow e^{x} = x \Rightarrow n o r e a l s o l u t i o n!

Answered by mr W last updated on 11/Jan/21

e^{x} = x \in C

\Rightarrow x = \ln x

x = {r e}^{i θ} = \ln r + i θ = r \cos θ + r \sin θ i

\ln r = r \cos θ \Rightarrow r = e^{r \cos θ}

θ = r \sin θ

\Rightarrow r = \frac{θ}{\sin θ} \dots (I)

\Rightarrow \frac{θ}{\sin θ} = e^{\frac{θ}{\tan θ}} \dots (I I)

(θ, r) = (\pm 1.3372, 1.3746), (\pm 7.5886, 7.8639), \dots

t h e r e a r e i n f i n i t e s o l u t i o n s .

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