How Do Different Definitions Impact the Infinite Power Tower's Domain?

[Dschumanji]

I have recently been studying the infinite power tower:

$f(x) = x \uparrow\uparrow\infty$

The function should actually be written with the infinity replaced by an n and the whole expression evaluated as a limit as n goes to infinity, but I am terrible with Latex. Anyways, I noticed that it is possible to redefine the function recursively:

$f(x) = x^{f(x)}$

It is possible to use this recursive definition to redefine f(x) as an explicit function of x:

$f(x) = \frac{W_{0}(-ln(x))}{-ln(x)}$

Where $W_{0}$ is the main branch of the Lambert W Function. Now, I read that Euler proved that the infinite power tower is only defined over the interval $[e^{-e}, e^{e^{-1}}]$. The first alternate definition of the infinite power tower is defined over the interval $(0, e^{e^{-1}}]$. The second alternate definition is defined over the interval $(0, 1)\cup(1, e^{e^{-1}}]$.

I have a large number of questions:

Do these differences in the domains of the first and second alternate definitions of the infinite power tower mean that they are not equal to the original definition of the infinite power tower? If this is the case, why are mathematicians justified in using the first and second alternate definitions to deduce the upper limit of the domain for the infinite power tower? How are mathematicians justified in using the second alternate definition as a means of computing the infinite power tower for complex numbers? Are the two alternate definitions not really definitions but statements about what properties the values in the domain and range of the infinite power tower must satisfy? How did I lose information by rewriting the infinite power tower recursively instead of as a limit? Does anyone have any idea how Euler proved the lower limit of the domain for the infinite power tower?

 

[disregardthat]

You can't define a function from the reals to the reals recursively, what you mean is defining it implicitly, by saying that f is the function such that f(x) = x^f(x). This "definition" ignores that the domain is not specified, and assumes that if there are solutions on some domain, they are unique.



A far better way is considering the sequence a_(n+1) = b^(a_n), where a_0 = b, and try to find those b for which this sequence converges. That would be equivalent to find the maximal domain for such a power-tower. Euler proved that the sequence defined above converges for b in [e^(-e),e^(e^(-1))].

Your rewriting of the implicitly defined function would be valid, but only wherever the power tower actually converges. Even though the rewritten form would have a larger domain, it doesn't mean that this domain is valid for the power tower itself.

Compare:
1 + x + x^2 + x^3 + ... = 1/(1-x) for -1<x<1. However 1/(1-x) is defined for x = 2, so does that mean 1 + 2 + 2^2 + 2^3 + ... = 1/(1-2) = -1? Of course it does not, the limit of that series does not converge, even though the alternate form has a larger possible domain that the series itself.

 

[Dschumanji]

You can't define a function from the reals to the reals recursively

Why not?

 

[disregardthat]

Why not?

By your example of a recursive definition, you actually mean an implicit definition, which is problematic due to the reasons given above. Please read what a recursive definition is, so that you'll understand why it doesn't make sense to define a real function recursively. (Unless the subject was transfinite recursion, which is something completely different, but not very relevant here)