01/25/2008, 11:36 PM
Dear Ivars!
Actually, I think that we should refer to a nice heart-like plot, element of a series made by Gottfried and re-visited by Andrews, where h, indicated as t = b^t is shown for b real and t complex, if I recall it correctly. The plot shows, in point t = + i (coordinates 0, +i, in the complex "t" plane, equivalent to complex number t = 0 + i*t) that the corresponding "base" parameter is b = 4. 810477381.. . Let call "r" this specific individual real number and try and solve the equation:
r ^ x = x, with, as we said, r = e^(Pi/2) = 4. 810477381..
Obviously, since we have: e^((Pi/2)*i) = i , but also e^((Pi/2)*(-i)) = -i , and since r is a real number > 1 (and therefore we must also have: r^(+oo )= +oo), the solutions of that equation are: x = {+oo, -i, +i}.
In fact, we have:
r ^ (-i) = -i
r ^ (+i) = +i
r ^ (+oo) = +oo
In fact, the Gottfried heart-like plot for real bases b > e^(1/e) shows an intersection with the imaginary axis (Real t = 0) in two symmetrical points, where the base parameter must be, in my opinion, the same: b = r.
In all the three above-mentioned solution cases, y = r ^ y ( or t = r ^ t) is equivalent to y = r # +oo, the infinite tetrate of r. We could then write that, in y = b # x , we might have:
y = b # +oo = -i )
y = b # +oo = +i ) for b = r
y = b # +oo = +oo )
The third line can be well accepted by everybody. For the first two lines there is a big problem of understanding (an infinite tetration, for b > e^(1/e), can be imaginary?). This is the ... terrific fact that is attracting all our attention. A mysterious (just to quote Henryk) link is connecting the infinity with the imaginary unit, as a condequence of ... tetration. I presume that this is also what is really bothering you! I shall try to post a note for explaining what I really mean, but I don't expect to be supported by everybody. We shall see!
To understand all this is very difficult. Nevertheless, please, let us try to avoid to differentiate constants!
(; -> )>>>
GFR
Actually, I think that we should refer to a nice heart-like plot, element of a series made by Gottfried and re-visited by Andrews, where h, indicated as t = b^t is shown for b real and t complex, if I recall it correctly. The plot shows, in point t = + i (coordinates 0, +i, in the complex "t" plane, equivalent to complex number t = 0 + i*t) that the corresponding "base" parameter is b = 4. 810477381.. . Let call "r" this specific individual real number and try and solve the equation:
r ^ x = x, with, as we said, r = e^(Pi/2) = 4. 810477381..
Obviously, since we have: e^((Pi/2)*i) = i , but also e^((Pi/2)*(-i)) = -i , and since r is a real number > 1 (and therefore we must also have: r^(+oo )= +oo), the solutions of that equation are: x = {+oo, -i, +i}.
In fact, we have:
r ^ (-i) = -i
r ^ (+i) = +i
r ^ (+oo) = +oo
In fact, the Gottfried heart-like plot for real bases b > e^(1/e) shows an intersection with the imaginary axis (Real t = 0) in two symmetrical points, where the base parameter must be, in my opinion, the same: b = r.
In all the three above-mentioned solution cases, y = r ^ y ( or t = r ^ t) is equivalent to y = r # +oo, the infinite tetrate of r. We could then write that, in y = b # x , we might have:
y = b # +oo = -i )
y = b # +oo = +i ) for b = r
y = b # +oo = +oo )
The third line can be well accepted by everybody. For the first two lines there is a big problem of understanding (an infinite tetration, for b > e^(1/e), can be imaginary?). This is the ... terrific fact that is attracting all our attention. A mysterious (just to quote Henryk) link is connecting the infinity with the imaginary unit, as a condequence of ... tetration. I presume that this is also what is really bothering you! I shall try to post a note for explaining what I really mean, but I don't expect to be supported by everybody. We shall see!
To understand all this is very difficult. Nevertheless, please, let us try to avoid to differentiate constants!
(; -> )>>>
GFR