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)

 

[CellCoree]

I am fascinated by the concept of the infinite power tower and the different ways in which it can be defined and evaluated. While I am not an expert in mathematics, I can provide some insights and offer a response to your questions.

Firstly, the differences in the domains of the first and second alternate definitions of the infinite power tower do not necessarily mean that they are not equal to the original definition. It is possible that the alternate definitions are equivalent to the original definition within their respective domains. However, it is important to note that the alternate definitions may not capture the full behavior of the original definition.

Mathematicians are justified in using the first and second alternate definitions to deduce the upper limit of the domain for the infinite power tower because they provide a useful approximation for the behavior of the original definition. These alternate definitions may not be exact, but they can still provide valuable insights and help in understanding the behavior of the infinite power tower.

Similarly, the second alternate definition can be used to compute the infinite power tower for complex numbers because it provides a convenient way to extend the original definition to a larger domain. This allows for a more comprehensive understanding of the infinite power tower and its behavior.

Regarding your question about the alternate definitions being statements about the properties of the infinite power tower, it is important to note that they are still valid definitions. They may not capture the full behavior of the original definition, but they can still be useful in certain contexts.

As for how Euler proved the lower limit of the domain for the infinite power tower, I am not familiar with the specific details of his proof. However, it is likely that he used mathematical techniques and reasoning to establish the lower limit.

Finally, rewriting the infinite power tower recursively instead of as a limit may result in the loss of some information. When we define a function recursively, we are essentially defining it in terms of itself, which can sometimes lead to a loss of information. On the other hand, defining a function as a limit allows for a more precise and comprehensive understanding of its behavior.

In conclusion, the infinite power tower is a fascinating concept that continues to be studied and explored by mathematicians and scientists. While there may be some differences in the alternate definitions and their domains, they can still provide valuable insights and understanding of the infinite power tower. I hope this response has helped to address some of your questions and provided some insights into this intriguing topic.