09/02/2007, 09:46 AM
Hi everybody!
Now, Enryk will certainly say that I am again late in reacting. He is right! Sorry about that. I shall improve! I still have to learn how to swim in this Forum and I really hate the TeX language, unless I am able to find an easy compiler that will help me.
Coming back to the subject, I more or less agree with all the previous comments (including the expressed doubts), keeping in mind that we may say that y = b^x for x<0 does not exist, only if we decide to remain in the "reality". If we admit the "complex universe", this is no more true. Complex analytic functions would probably help us in the analysis of the entire b>0 domain, for y = b # x (b real). Unfortunately, I am not very "fluent" in that area.
Therefore, as I promised to Enryk, I submit to your attention the attached pdf notes, hoping that you will not be annoyed and that find some ideas to be developed. Please see also what should, in my opinion, happen in the range 0 < b < e^(-e). It is a simulation but, just tell me what you think. In fact, I agree that that interval corresponds to the negative bases, for exponentiation. The appearence of oscillations, for b<1 must be admitted, if we accept to go out of the "reality". I think to understand a little bit better what Euler meant by saying that the "infinite towers" don't converge
for b < e ^(-e). Perhaps, they just oscillate -> oo.
Please also investigate, in the 1 < b < e^(1/e) interval, what could be the role of the second (upper) "unreachable" asymptote.
Another fact which I should like to mention is that the superlog (slog) should be limited to b > e^(1/e) [the jayfox eta, or the self-root of e], otherwise ... we are in trouble (apart from any possible theoretical assessment).
Congratulations to Andrew. Well done! But ... the war continues!
Rubtsov and myself, we are investigating also in other directions. We shall keep you informed asap.
Thank you
G. F. Romerio
Now, Enryk will certainly say that I am again late in reacting. He is right! Sorry about that. I shall improve! I still have to learn how to swim in this Forum and I really hate the TeX language, unless I am able to find an easy compiler that will help me.
Coming back to the subject, I more or less agree with all the previous comments (including the expressed doubts), keeping in mind that we may say that y = b^x for x<0 does not exist, only if we decide to remain in the "reality". If we admit the "complex universe", this is no more true. Complex analytic functions would probably help us in the analysis of the entire b>0 domain, for y = b # x (b real). Unfortunately, I am not very "fluent" in that area.
Therefore, as I promised to Enryk, I submit to your attention the attached pdf notes, hoping that you will not be annoyed and that find some ideas to be developed. Please see also what should, in my opinion, happen in the range 0 < b < e^(-e). It is a simulation but, just tell me what you think. In fact, I agree that that interval corresponds to the negative bases, for exponentiation. The appearence of oscillations, for b<1 must be admitted, if we accept to go out of the "reality". I think to understand a little bit better what Euler meant by saying that the "infinite towers" don't converge
for b < e ^(-e). Perhaps, they just oscillate -> oo.
Please also investigate, in the 1 < b < e^(1/e) interval, what could be the role of the second (upper) "unreachable" asymptote.
Another fact which I should like to mention is that the superlog (slog) should be limited to b > e^(1/e) [the jayfox eta, or the self-root of e], otherwise ... we are in trouble (apart from any possible theoretical assessment).
Congratulations to Andrew. Well done! But ... the war continues!
Rubtsov and myself, we are investigating also in other directions. We shall keep you informed asap.
Thank you
G. F. Romerio