Oooops, sorry ... ! I changed my mind.
After a carefully re-thinking, allow me to come back, shortly to my conjecture, which I propose to formulate as follows:
"In the domain b >= e ^ (1/e) [b >= Eta], the plot of the real branch of y = b # x [b-tetra-x] show a smooth one-valued always increasing real function. Its first derivative is always >= 0 and its second derivative is a real function with a zero at x = xf. The value of xf against b is a smooth always decreasing function, with a horizontal asymptote for b -> + oo, at x = -1 and a vertical asymptote for b -> Eta, wher x = + oo". But, Andydude said:
After the examination of the attached Andrew's plot (Andydude first test) , I think that the shown approximations seem not appropriate enought. Concerning my erratic behaviour, Father Euler would say: "Errare humanum est, perseverare diabolicum". Henryk would probably say: " It is not necessary to be crazy to be here, but it helps !!". Andydude will be patient, I hope!
The problem is that the perturbations of the Andydude tests in the upper-left corner of his figure (lower bases) are strange and probably due to computing overflow. In fact, the reptilian part of my brain and my ... religious beliefs, together with the simulation of y = b# x for b = Eta = e^(1/e) = 1.44456..., assure me that y -> e for x -> +oo, with an always increasing behaviour. Both the y first and second derivatives seem to become zero at x -> +oo, when we shall have y = e. This should mean that for b = Eta, we must have y = e and y" = y' = 0 at x = + oo. Either Andrew's derivation procedures were not accurate enought, or they went overflow or the function itself to which they were applied is not sufficiently appropriate, for low bases.
Please see the "GFR qualitative draft", in annex. For b < Eta we are in trouble, for several reasons (multiple and/or complex values).
Maybe, next time!
@IVARS. I haven't studied your developments concerning hunting for a possible "mother" differential equation. For the moment I leave the possibility to interact to other Participants. I'll be back on that soon. I think it is a very important issue.
GFR
After a carefully re-thinking, allow me to come back, shortly to my conjecture, which I propose to formulate as follows:
"In the domain b >= e ^ (1/e) [b >= Eta], the plot of the real branch of y = b # x [b-tetra-x] show a smooth one-valued always increasing real function. Its first derivative is always >= 0 and its second derivative is a real function with a zero at x = xf. The value of xf against b is a smooth always decreasing function, with a horizontal asymptote for b -> + oo, at x = -1 and a vertical asymptote for b -> Eta, wher x = + oo". But, Andydude said:
andydude Wrote:Well, I have constructed a function to find the values that GFR is looking for. The function is defined such that: \( y''(f(b)) = 0 \) where \( y(x) = {}^{x}b \). I have attached a plot of this function for the first few approximations (n=5..9) of tetraiton using the natural/inverse-slog tetration method. It looks as though this function is very well-behaved for \( b > e \), but is quite slow to converge for lower bases. So my guess is that GFR's conjecture that f(2)=0 is probably not true.
After the examination of the attached Andrew's plot (Andydude first test) , I think that the shown approximations seem not appropriate enought. Concerning my erratic behaviour, Father Euler would say: "Errare humanum est, perseverare diabolicum". Henryk would probably say: " It is not necessary to be crazy to be here, but it helps !!". Andydude will be patient, I hope!
The problem is that the perturbations of the Andydude tests in the upper-left corner of his figure (lower bases) are strange and probably due to computing overflow. In fact, the reptilian part of my brain and my ... religious beliefs, together with the simulation of y = b# x for b = Eta = e^(1/e) = 1.44456..., assure me that y -> e for x -> +oo, with an always increasing behaviour. Both the y first and second derivatives seem to become zero at x -> +oo, when we shall have y = e. This should mean that for b = Eta, we must have y = e and y" = y' = 0 at x = + oo. Either Andrew's derivation procedures were not accurate enought, or they went overflow or the function itself to which they were applied is not sufficiently appropriate, for low bases.
Please see the "GFR qualitative draft", in annex. For b < Eta we are in trouble, for several reasons (multiple and/or complex values).
Maybe, next time!
@IVARS. I haven't studied your developments concerning hunting for a possible "mother" differential equation. For the moment I leave the possibility to interact to other Participants. I'll be back on that soon. I think it is a very important issue.
GFR
