Range of complex tetration as real
#1
Running my tetration code I was reminded that my results for $$^x a \textrm{, where } 1<a<1.444$$ have their range on the real line. Do other extensions of tetration work in a similar fashion?
Daniel
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#2
(10/13/2022, 05:11 AM)Daniel Wrote: Running my tetration code I was reminded that my results for $$^x a \textrm{, where } 1<a<1.444$$ have their range on the real line. Do other extensions of tetration work in a similar fashion?

As far as I remember the Paulsen continuation has a super small imaginary part on the real axis for those bases. But for those bases there are not much different from regular tetrations out there. The interesting bases are surely $$b>e^{1/e}$$ and there is much more variety with different methods.
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#3
(10/13/2022, 05:11 AM)Daniel Wrote: Running my tetration code I was reminded that my results for $$^x a \textrm{, where } 1<a<1.444$$ have their range on the real line. Do other extensions of tetration work in a similar fashion?

With my extention, as long as you're not doing any tetration towards -2 or less, it always gives real outputs with real heights, ONLY if the base is higher than 1. Otherwise my extension do gives complex results.
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