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Challenging the concept of real tetration
#1
So Kneser derives the real half iterate of the exponential function. I think his proof is great, but he makes use of the complex numbers to obtain his results on the real numbers. Shouldn't a solution on the real numbers have a proof not resorting to complex numbers?
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#2
(07/04/2022, 09:56 AM)Daniel Wrote: So Kneser derives the real half iterate of the exponential function. I think his proof is great, but he makes use of the complex numbers to obtain his results on the real numbers. Shouldn't a solution on the real numbers have a proof not resorting to complex numbers?

I dont wanna be rude or annoying but most methods here are not based on complex numbers or complex fixpoints.
We make complex plots to convince ourselves that they are analytic though.

Although most are not proven and such, I find it remarkable that you ask this.

Also simply using the appropriate analytic 1-periodic function theta(z) transforms any analytic tetration to another analytic tetration; by using the simple

tetrationnew(z) = tetrationold(z + theta(z))

In essense even the riemann mapping/kneser mapping is a theta thing.

I recently added my personal ( subjective ?) list of types of ways to do tetration.

most methods (like kneser too ) do not solve directly for a half-iterate ( as you mentioned in the OP ) but instead construct a superfunction or abel function , sexp or slog.

Since ln(z) does not have real fixpoints , it makes sense that many fixpoints methods start from a complex fixpoint.

Also we usually want some kind of uniqueness criterion, although in the last few years we somewhat did not focus on it imo.

Apart from analytic methods there are also C^oo methods.
But the consensus is that analytic is nicer.

I do not see what there " is to challange ".
I see no contraditions.

And i suspect most here feel the same way.


regards

tommy1729
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#3
(07/04/2022, 12:38 PM)tommy1729 Wrote:
(07/04/2022, 09:56 AM)Daniel Wrote: So Kneser derives the real half iterate of the exponential function. I think his proof is great, but he makes use of the complex numbers to obtain his results on the real numbers. Shouldn't a solution on the real numbers have a proof not resorting to complex numbers?

I dont wanna be rude or annoying but most methods here are not based on complex numbers or complex fixpoints.
We make complex plots to convince ourselves that they are analytic though.

Although most are not proven and such, I find it remarkable that you ask this.

Also simply using the appropriate analytic 1-periodic function theta(z) transforms any analytic tetration to another analytic tetration; by using the simple

tetrationnew(z) = tetrationold(z + theta(z))

In essense even the riemann mapping/kneser mapping is a theta thing.

I recently added my personal ( subjective ?) list of types of ways to do tetration.

most methods (like kneser too ) do not solve directly for a half-iterate ( as you mentioned in the OP ) but instead construct a superfunction or abel function , sexp or slog.

Since ln(z) does not have real fixpoints , it makes sense that many fixpoints methods start from a complex fixpoint.

Also we usually want some kind of uniqueness criterion, although in the last few years we somewhat did not focus on it imo.

Apart from analytic methods there are also C^oo methods.
But the consensus is that analytic is nicer.

I do not see what there " is to challange ".
I see no contraditions.

And i suspect most here feel the same way.


regards

tommy1729

Thanks Tommy, I think it is cool that folks have proofs that only use the real numbers.
Daniel

Edit: I apologize for the title. I'm experimenting with stating things in a controversial manner so as to get a response.
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