10/03/2007, 11:25 AM
GFR Wrote: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).
Though metaphysically this thought perhaps makes sense, we have to admit that by mathematical consideration
\( \lim_{n\to\infty} {^nb}=\infty \) for \( b>\eta \).
Quote:Actually, it is well known that: e^(Pi/2) = i.Hey guys, dont forget the \( i \) in the exponent!
\( e^{\frac{\pi}{2}i}=i \).
Quote:This means that an infinite tower x = b#oo, with b = 4.810477381.., may have an imaginary "height" of x = i = +/- sqrt(-1).
There can at most be one limit however there are many fixed points.
Quote: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?
No, no, no.
