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Complex to real tetration via Kneser
#1
Can the Riemann mapping theorem be used to convert a complex solution for tetration to a real solution? If so then the solutions for Schroeder's functional equation on my website Tetration.org can be mapped into real solutions. Since the mapping is biholomorphic then real solutions could be mapped into complex ones.
Daniel
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#2
im not sure what you are asking.

the kneser method has a riemann mapping making it real valued tetration for its real base larger than eta ( e^(1/e) )

so that is complex going to real.

IF instead you meant complex bases , well then do you want to use a kind of base change turning them into real bases ??

and why not directly using the real bases ?

Or do you want the complex bases to give a real valued tetration ? that would not make sense imo so I guess not.


I think the gaussian method works best for base change ;

f_b(s+1) = exp( ln(b) t(s) f_b(s) )

This seems analytic in the base b.

Using that f_b in the usual way to get the gaussian method , I think will preserve analytic in the base b.

I know , I know , promoting my own ideas again , but still I believe that.

I see no reason why not.


regards

tommy1729
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#3
(07/01/2022, 03:55 AM)Daniel Wrote: Can the Riemann mapping theorem be used to convert a complex solution for tetration to a real solution? If so then the solutions for Schroeder's functional equation on my website Tetration.org can be mapped into real solutions. Since the mapping is biholomorphic then real solutions could be mapped into complex ones.

Depends what you are asking. Kneser explicitly does this procedure. If you mean for complex bases, I suggest Paulsen's paper. He describes how to analytically continue the Kneser mapping theorem from \(b > \eta\) to every base except \(0\), with a branch cut along the negative axis, (iirc).

It does use Schroder's functional equation about two fixed point pairs, \(L^+,L^-\), where for real \(b > \eta\) these are complex conjugates.
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#4
Why care if we can easily move between real and complex tetation? Well intellectually, it would allow the removal of the dichotomy between real and complex. I could mesh my own research on complex tetration with the research on the Tetration Forum, mix and match theorems and techniques.
Daniel
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