Concerning:
(1) - General observations
The study of the fixpoints of y =f(x) [... simply ...!] means the analysis of the points where f(x) = x, i.e. the intersections of y = f(x) with y = x, the principal diagonal of the xy plane. Fixpoint analysis is an excellent tool used by modern mathematicians for finding serial developments and, particularly, by specialists in analytic functions, recursivists, iterationists, fractalists, as well as experts in AI and NKS matters. The application of this tool to our subject implies the study of the intersections of y = b^x with y = x, for various b, i.e. for finding the values of x for which b^x = x.
At the same time, b^x = x (i.e. b^x - x = 0) can be seen as an implicit functional equation, the solutions of which should represent, after infinite iterations, an infinite tetration (infinite tetrates) of base b, which can be written as : y = b#(+oo) = h, the heights of the "infinite tower" with base b.
(2) - Some calculations attempts
Some months ago, a friend of mine asked me, for fun, to estimate the numerical value of 7#3 = 7-tetra-3 = 7^(7^7). My pocket calculatort immediately gave me 7^823543 and, after that, it went to overflow. Then I decided to use Mathematica and I got:
7#3 = 3.759823526783788538...x 10^695974 = 3759........2343, an integer number with 695975 figures, covering a printout of about 87 DIN A4 pages.
Therefore, yesterday, I decided to follow a lower profile and tried with a more reasonable and "famous" base, i.e.: rho = 4.810477381... . Well, always using Mathematica, I got:
rho#3 = 6.8101069808199648...X10^1304, which is much more civilized. It is not an integer number, but a DIN A4 page will be enought to show it, with a reasonable precision.
Then, this morning, after my breakfast, I tried to imagine how large could be rho#4 and I had to drink four cups of coffee, to recover. Now (perseverare ... diabolicum) I cannot avoid thinking of what could be rho#100 or rho#1000 or, even, [n->+oo]lim (rho#n) !
Why (... the Hell!), at the same time, equation rho^x = x gives as solution x = {-i, +i} ? Can an infinite tetration of base "rho" softly collapse to a conjugate imaginary unit ? This, of course, while in expression such as y = rho^n superexponent n "goes" to infinity?
(2) - Kritik of the Mathematical Reason
It is expected that the critical point will be to say that one thing is the limit of rho^n, for n-> oo, and another thing is the determination of the fixpoint of rho^x = x. The countercritical position could be to say that, in this particular case, the two procedures must reach the same non-contradictory results (divine surprise!). Then, our position could be:
- to say that they don't;
- to say that, unfortunately, they do (reach non-contradictory results).
The fact is that, if they do bring us to coherent results, the infinite tetration of base rho (sorry, this is a nightmare of mine) should give: rho#oo = {-i, +i, +oo}. As a matter of fact, in the domain of b > e^(1/e), the "real" solution +oo is (at least, thoretically) always obtainable by an infinite number of iterative calculations and complex solutions are calculable via the inversion of b = y^(1/y), i.e.: y = b^y. The apparent overlapping of the two strategies is that also +oo seems to be a fixpoint of y = b^y , in fact b^(+oo) = +oo. But it is not quite so, since b^(+oo) is of an infinite order much larger than +oo. In other words, for any b > eta [eta = e^(1/e)], b^x >> x, for x -> +oo, the real plots of y = b^x and y = x will never cross eachother for x -> +oo (I mean, before x = +oo). So, what! They will cross ... after ?? But, after what?! After ... (I don't dare to say) ... infinite? I can't believe it. That would mean a "mathematical fiction" relation of ... order between the real and the complex numbers. Now, Henryk, you are authorized to say: "No, No, No!". Perhaps, I am joking!
In conclusion, what shall we do? Forget the relation of order. Please read the attached pdf notes and tell me, if you wish, your reactions. They also contain the plots provided by Gottfried and Andydude. Or, please destroy them ... before reading, if you prefer so.
Forgive me for any mistake, omission or lack of precision and remember: "Also Human Ignorance (I mean mine) is a Gift of God!". (Otto von Bismarck, I presume).
Thanks!
GFR
bo198214 Wrote:There is also a mysterious base where the primary fixed point is exactly \( i \). This is the case for \( i=b^i \) which is clearly satisfied for \( b=e^{\frac{\pi}{2}}\approx 4.8104 \).Some comments:
(1) - General observations
The study of the fixpoints of y =f(x) [... simply ...!] means the analysis of the points where f(x) = x, i.e. the intersections of y = f(x) with y = x, the principal diagonal of the xy plane. Fixpoint analysis is an excellent tool used by modern mathematicians for finding serial developments and, particularly, by specialists in analytic functions, recursivists, iterationists, fractalists, as well as experts in AI and NKS matters. The application of this tool to our subject implies the study of the intersections of y = b^x with y = x, for various b, i.e. for finding the values of x for which b^x = x.
At the same time, b^x = x (i.e. b^x - x = 0) can be seen as an implicit functional equation, the solutions of which should represent, after infinite iterations, an infinite tetration (infinite tetrates) of base b, which can be written as : y = b#(+oo) = h, the heights of the "infinite tower" with base b.
(2) - Some calculations attempts
Some months ago, a friend of mine asked me, for fun, to estimate the numerical value of 7#3 = 7-tetra-3 = 7^(7^7). My pocket calculatort immediately gave me 7^823543 and, after that, it went to overflow. Then I decided to use Mathematica and I got:
7#3 = 3.759823526783788538...x 10^695974 = 3759........2343, an integer number with 695975 figures, covering a printout of about 87 DIN A4 pages.
Therefore, yesterday, I decided to follow a lower profile and tried with a more reasonable and "famous" base, i.e.: rho = 4.810477381... . Well, always using Mathematica, I got:
rho#3 = 6.8101069808199648...X10^1304, which is much more civilized. It is not an integer number, but a DIN A4 page will be enought to show it, with a reasonable precision.
Then, this morning, after my breakfast, I tried to imagine how large could be rho#4 and I had to drink four cups of coffee, to recover. Now (perseverare ... diabolicum) I cannot avoid thinking of what could be rho#100 or rho#1000 or, even, [n->+oo]lim (rho#n) !
Why (... the Hell!), at the same time, equation rho^x = x gives as solution x = {-i, +i} ? Can an infinite tetration of base "rho" softly collapse to a conjugate imaginary unit ? This, of course, while in expression such as y = rho^n superexponent n "goes" to infinity?
(2) - Kritik of the Mathematical Reason
It is expected that the critical point will be to say that one thing is the limit of rho^n, for n-> oo, and another thing is the determination of the fixpoint of rho^x = x. The countercritical position could be to say that, in this particular case, the two procedures must reach the same non-contradictory results (divine surprise!). Then, our position could be:
- to say that they don't;
- to say that, unfortunately, they do (reach non-contradictory results).
The fact is that, if they do bring us to coherent results, the infinite tetration of base rho (sorry, this is a nightmare of mine) should give: rho#oo = {-i, +i, +oo}. As a matter of fact, in the domain of b > e^(1/e), the "real" solution +oo is (at least, thoretically) always obtainable by an infinite number of iterative calculations and complex solutions are calculable via the inversion of b = y^(1/y), i.e.: y = b^y. The apparent overlapping of the two strategies is that also +oo seems to be a fixpoint of y = b^y , in fact b^(+oo) = +oo. But it is not quite so, since b^(+oo) is of an infinite order much larger than +oo. In other words, for any b > eta [eta = e^(1/e)], b^x >> x, for x -> +oo, the real plots of y = b^x and y = x will never cross eachother for x -> +oo (I mean, before x = +oo). So, what! They will cross ... after ?? But, after what?! After ... (I don't dare to say) ... infinite? I can't believe it. That would mean a "mathematical fiction" relation of ... order between the real and the complex numbers. Now, Henryk, you are authorized to say: "No, No, No!". Perhaps, I am joking!
In conclusion, what shall we do? Forget the relation of order. Please read the attached pdf notes and tell me, if you wish, your reactions. They also contain the plots provided by Gottfried and Andydude. Or, please destroy them ... before reading, if you prefer so.
Forgive me for any mistake, omission or lack of precision and remember: "Also Human Ignorance (I mean mine) is a Gift of God!". (Otto von Bismarck, I presume).
Thanks!
GFR
