This was something I had fun uncovering with a friend of mine. Others had already discovered it, but sometimes its fun to work out examples like this because they're actually fairly surprising at time.

It started with me exploring iterated functions on a calculator, in this case, simple applying the power recursively.

At first I'd though there would only be one trivial case z where this had a solution.

However I discovered to my surprise that there was another,

And so I started thinking why this works on computers. My intuition seemed to tell me that the number should be growing finitely for all z larger than 1. So I went back to studying what happens as you make this tower of powers progressively higher for various values of z.

There were two problems to be solved. The first one was within what range of z was this limit defined, and when what was the limit when it was defined. It turned out that the latter problem was easier to solve than the first, and surprisingly that an analytical solution actually existed, using Lambert's W function!

This function is defined as

This function is fast and easy to compute using various algorithms. It has two principal branches in the real space, but if we follow its principal branch its defined from 1/e to infinity. It can be used kinda like an expanded logarithm. Like so.

Using that its straightforward to derive a closed expression for the analytical function we're interested in.

Now that assumes of course that the limit does in fact exist, but when it does it has this neat symmetric form.

It started with me exploring iterated functions on a calculator, in this case, simple applying the power recursively.

At first I'd though there would only be one trivial case z where this had a solution.

However I discovered to my surprise that there was another,

And so I started thinking why this works on computers. My intuition seemed to tell me that the number should be growing finitely for all z larger than 1. So I went back to studying what happens as you make this tower of powers progressively higher for various values of z.

There were two problems to be solved. The first one was within what range of z was this limit defined, and when what was the limit when it was defined. It turned out that the latter problem was easier to solve than the first, and surprisingly that an analytical solution actually existed, using Lambert's W function!

This function is defined as

This function is fast and easy to compute using various algorithms. It has two principal branches in the real space, but if we follow its principal branch its defined from 1/e to infinity. It can be used kinda like an expanded logarithm. Like so.

Using that its straightforward to derive a closed expression for the analytical function we're interested in.

Now that assumes of course that the limit does in fact exist, but when it does it has this neat symmetric form.

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