I have recently been studying the infinite power tower:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]f(x) = x \uparrow\uparrow\infty[/itex]

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:

[itex]f(x) = x^{f(x)}[/itex]

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

[itex]f(x) = \frac{W_{0}(-ln(x))}{-ln(x)}[/itex]

Where [itex]W_{0}[/itex] 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 [itex][e^{-e}, e^{e^{-1}}][/itex]. The first alternate definition of the infinite power tower is defined over the interval [itex](0, e^{e^{-1}}][/itex]. The second alternate definition is defined over the interval [itex](0, 1)\cup(1, e^{e^{-1}}][/itex].

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?

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# The Infinite Power Tower

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