e^e^x = x x = ?


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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

$e^{e^{. . . e^{x}}} = x \Leftrightarrow e^{x} = x \Rightarrow n o r e a l s o l u t i o n!$
	Answered by mr W last updated on 11/Jan/21

	$e^{x} = x \in C$ $\Rightarrow x = \ln x$ $x = {r e}^{i θ} = \ln r + i θ = r \cos θ + r \sin θ i$ $\ln r = r \cos θ \Rightarrow r = e^{r \cos θ}$ $θ = r \sin θ$ $\Rightarrow r = \frac{θ}{\sin θ} . . . (I)$ $\Rightarrow \frac{θ}{\sin θ} = e^{\frac{θ}{\tan θ}} . . . (I I)$ $(θ, r) = (\pm 1.3372, 1.3746), (\pm 7.5886, 7.8639), . . .$ $t h e r e a r e i n f i n i t e s o l u t i o n s .$
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