Kouznetsov Wrote:By the way, your deduction gives the hint, how to prove the Theorem 1.
Perhaps. I am not convinced yet that it is true, but if it was, this would be great. We even know vice versa if we have two solutions \( F \) and \( G \) of the Abel equation then \( G^{-1}(F(x+1))-(x+1)=G^{-1}(\exp(F(x)))-(x+1)=G^{-1}(\exp(G(G^{-1}(F(x)))))-(x+1)=G^{-1}(F(x))+1-(x+1)=G^{-1}(F(x))-x \), meaning that \( \phi=G^{-1}\circ F-\text{id} \) is a 1-periodic function, so we know already that each other solution \( G \) of the Abel equation must be of the form \( F(z+\phi(z)) \) for some 1-periodic \( \phi \). To prove theorem 1 everything depends on the behaviour of those 1-periodic functions for \( \Im(z)\to\infty \). For uniqueness roughly the real value must be go to infinity for the imaginary argument going to infinity.
Quote:(However, we have to scale the argument of sin function.)
Corrected in the original post.
Quote: How about the collaboration?That would be great.
Quote:(P.S. Does the number of replies I should answer grow as the Ackermann function of time, or just exponentially?)
I would guess its rather logarithmically (much questions at the start but then slowly ebbing away)! At least here is another question: Can you please compute values on the real axis for bases \( b<e^{1/e} \) for example for \( b=\sqrt{2} \)? I would like to compare your solution with the regular tetration developed at the lower real fixed point (which would be 2 in the case of \( b=\sqrt{2} \)) of \( b^x \).
@Andrew
Can you post a comparison graph with your slog/sexp? The values are in Dmitrii's paper. How does the periodicity of your slog (or was it sexp?) at the imaginary axis compare with Dmitrii's limit \( \lim_{y\to\infty} F(x+iy)=L \) (where \( L \) is a fixed point of \( \exp \))?