Kneser's tetration was conjectured to have no singularities in the upper half plane ( for Im(z) > 0 ).
Or was that Kouznetsov ? If I recall correctly ; both ... implying they are identical and unique.
I could be wrong though.
After all these years, I see little public info about singularities. ( apart from trivial log singularities at expected places )
Not that it is easy though. For a taylor series we try to prove the radius of convergeance for a function expansion somewhere... usually by special properties or by patterns and asymptotics in the n th derivatives.
But usually we do not get a nice taylor series with proven trends.
We get something for which no efficient radius of convergeance method is known or is efficient.
Converting to a taylor series often does not help immediately and feels like a restatement of the problem or a nonconstructive circular logic.
Apart from uniqueness by singularities there is the idea of uniqueness by bounds.
( sometimes proven, sometimes conjectures ... im not even sure without thinking about it first )
Those are just my impressions though and I could be wrong.
Also I just scratched the surface of the large amount of things that could have been said about it.
regards
tommy1729
Or was that Kouznetsov ? If I recall correctly ; both ... implying they are identical and unique.
I could be wrong though.
After all these years, I see little public info about singularities. ( apart from trivial log singularities at expected places )
Not that it is easy though. For a taylor series we try to prove the radius of convergeance for a function expansion somewhere... usually by special properties or by patterns and asymptotics in the n th derivatives.
But usually we do not get a nice taylor series with proven trends.
We get something for which no efficient radius of convergeance method is known or is efficient.
Converting to a taylor series often does not help immediately and feels like a restatement of the problem or a nonconstructive circular logic.
Apart from uniqueness by singularities there is the idea of uniqueness by bounds.
( sometimes proven, sometimes conjectures ... im not even sure without thinking about it first )
Those are just my impressions though and I could be wrong.
Also I just scratched the surface of the large amount of things that could have been said about it.
regards
tommy1729