10/03/2007, 11:06 AM
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 \( b=e^{\frac{\pi}{2}}\approx 4.8104 \).
The fixed points of functional equation x = b^x define, at the same time, the "heights" of the corresponding infinite towers: x = b#oo.
We have seen that, for bases b < e^(1/e), there are real x solutions. For b > e^(1/e), our discussion (see Gottfried, for example) seems to bring us to agree on complex "fixed points" satisfying x = b^x, for x real >0. This means that we may have infinite towers with complex values (heights). In particular, there is the mysterious fact shown by Henryk:
x = b^x, satisfied by x=i, if b = e^(Pi/2). Actually, it is well known that: e^(Pi/2) = i.
This means that an infinite tower x = b#oo, with b = 4.810477381.., may have an imaginary "height" of x = i = +/- sqrt(-1). This is also corroborated by the recent plots shown by Gottfried and by a plot I got using the "Mathematica" product-logarithm operators [order 0 and -1]. By the way, very beautiful animated plot, Henryk!
Are then we authorized to write: (4.810477381..)#oo = {-i, +i, +oo} ? Can we also see other complex "heights" (branches) ?
In other words, for b > Eta, together with a set of complex "heights" solutions, shall we always be authorized (obliged?) to also admit at least one infinite tower with infinite height?
Sorry to be so annoying and ... flat.
GFR
