0

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.

Blue
  • 72,688
  • 11
  • 116
  • 225
Joey Peluka
  • 69
  • 3
  • 3
    See 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
  • Thanks, but I don’t see how any of these are the same question. – Joey Peluka Dec 11 '19 at 00:47
  • 1
    The first link tells you that this is called an infinite tetration, how it is computed, and where it converges. The [wikipedia entry on tetration](https://en.wikipedia.org/wiki/Tetration#Infinite_heights) has a graph over the real and over the complex numbers. – Randy Marsh Dec 11 '19 at 00:56
  • 1
    Can't we write this as $y=x^y$ – Naman Jain Dec 11 '19 at 06:38

2 Answers2

0

Well, $y=x^{x^{x^{x^{...}}}}, y=x^y$

Using Wolfram the plot would look like this enter image description here

Naman Jain
  • 321
  • 1
  • 11
0

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
  • 241,289
  • 52
  • 105
  • 219