A Transfinite Taylor series of exp(x) and of h(x)

[H Tomasz Grzybowski]

Let h(h(x)) = exp(x), where h(⋅) is holomorphic in the whole ℂ plane.
I want an extension of the domain of exp(⋅) and of h(⋅) so that
we can find values of these functions for x = Aleph(0).

 

[jambaugh]

$\aleph_0^n = \aleph_0$ so the "Maclaurin series" of $\exp(\aleph_0)$ will be $\aleph_0$.
But this really isn't sensible since there is continuous measure on the space of transfinite cardinals. Or rather you will have to implement one to speak of limits of series and then your answer will depend on that choice.

Short of that you're using cardinal arithmetic. But notice the inconsistency in results. $2^{\aleph_0}=\aleph_1$ where we define the notation as a set of functions or power set, but
$e^{\aleph_0} \equiv 1 + \aleph_0 + (1/2!)\aleph_0^2 + \ldots = 1+\aleph_0 + \aleph_0 +\ldots = 1+\aleph_0\times\aleph_0 = \aleph_0$

 

[FactChecker]

I agree with @jambaugh . Furthermore, unless someone has laid the groundwork regarding a metric, continuity, convergence, etc. that includes $\aleph_0$, it is premature to consider the Taylor series. If such a groundwork and context has been laid, I am not aware of it.

 

[H Tomasz Grzybowski]

I clearly stated that I want TRANSFINITE Taylor series. When you look at a term omega+n of transfinite Taylor series, it follows from the Stirling formula that such a term evaluated at Aleph(0) wil have value Continuum. So will any transfinite sum not exceeding Continuum terms.

 

[H Tomasz Grzybowski]

The above refers to a transfinite Taylor series of exp(x). Taylor series of h(x) remains unknown.

 

[jambaugh]

I clearly stated that I want TRANSFINITE Taylor series. When you look at a term omega+n of transfinite Taylor series, it follows from the Stirling formula that such a term evaluated at Aleph(0) wil have value Continuum. So will any transfinite sum not exceeding Continuum terms.

You are speaking as if there is a standard definition of a Transfinite Taylor Series so would you be so kind to cite a reference. I've never seen such a definition (which is likely my failing but it certainly isn't a broadly known definition.) As to the Stirling formula I presume you're referring to the Stirling approximation of N! Is that correct? If so would you be so kind as to be a bit more explicit in your obvious "it follows from the Stirling formula" implication.

I'm familiar with the transfinite cardinals as well as the transfinite ordinals. And one can certainly define series in the abstract and the arithmetic thereof using series indexed by transfinite ordinals if you wish. You could even index with continuum variables and call them "integrals", and even index over function spaces if you pick some specific well ordering of that space. Basically the series are identified with the sequence of terms and one need not even insist on convergence.

But to speak of convergence in any context, namely to equate a series with a value in any sense you must apply some form of topology. What is the topology you are using when you speak of evaluating a Taylor series (transfinite or no) for the variable equaling a specific transfinite cardinal?

 

[H Tomasz Grzybowski]

Yes, by "Stirling formula" I mean the approximation of n!.
Regarding exp(Aleph(0)), each tansfinite term is equal to Continuum, so if there are no more than Continuum terms,
the sum equals Continuum.
My question is about possible extension of Taylor series of h(x), but as of now,
I do not know the usual Taylor series of it.