Image you had the graph where $x$ is raised to itself an infinite amount of times.
Like this: $$y=x^{x^{x^{x^\cdots}}}$$
What would this graph look like, and/or how is that computed?
Thanks in advance.
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3See https://math.stackexchange.com/q/661608/391136 and https://math.stackexchange.com/q/166433/391136 and https://math.stackexchange.com/q/492109/391136 – Randy Marsh Dec 10 '19 at 18:58
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Thanks, but I don’t see how any of these are the same question. – Joey Peluka Dec 11 '19 at 00:47
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1The first link tells you that this is called an infinite tetration, how it is computed, and where it converges. The wikipedia entry on tetration has a graph over the real and over the complex numbers. – Randy Marsh Dec 11 '19 at 00:56
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1Can't we write this as $y=x^y$ – Naman Jain Dec 11 '19 at 06:38
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The explicit solution of $y=x^y$ is given in terms of Lambert function $$y=-\frac{W(-\log (x))}{\log (x)}$$ and, in the real domain, it is bounded at $x=e^{\frac{1}{e}}$.
Naman Jain's answer gives the plot of it.
Claude Leibovici
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@JoeyPeluka. Because, in the real domain, $W(t)$ is defined if $t\geq -\frac 1e$. Have a look at https://en.wikipedia.org/wiki/Lambert_W_function – Claude Leibovici Dec 12 '19 at 04:00