(02/24/2010, 11:16 AM)bo198214 Wrote: Hm, interesting idea.
For \( b>e^{1/e} \) this line would however have a gap near the real axis?
I show a slightly modified type of plot. Here I use b=exp(1).
I found, that the complex fixpoints t_k can be parametrized in terms of their real part. Denote a fixpoint t_k = a + b*i, then I can select some a and compute the required b. Then I can also compute the number k by taking t_k- log(t_k).
It is more difficult to compute t_k by choosing k: we must iterate x = log(x) + 2PiI*k many times as usual, however the only possibility to find the fixpoints for exactly integer k.
In the plot the blue lines are the locus of the fixpoints assuming continuous real k. The fixpoints at integer k are marked with big red points; the "fixpoint" at k/2 is marked by a star.
The blue lines are naturally continuous *below* real(t_0)~0.3181... however, selecting any real k we cannot find that points. In fact, if we compute such a point using, for instance, a=0.1, then we get some meaningful b and the according k, but if we resinsert that k to compute the associated fixpoint we end in the curve with real parts *above* real(t_0). So I mared that "inaccessibl area" lefthand in the plot with a grey shade.
The orange lines are the computed (continuous) indexes k; their y-scale is the orange scale on the right side of the plot (the line has abscissa "1" at the fixpoint t_k at k=1 and abscissa "0" at fixpoint t_k at k=0 )
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[update]
I actually do not yet know how this can be useful - I've not yet used such a fixpoint-shift for the recentering of the exp-function with base e seriously, it was always somehow messy....
[/update]
Here is the plot:
Gottfried Helms, Kassel
