01/13/2016, 10:58 PM
(This post was last modified: 01/14/2016, 04:35 AM by sheldonison.)
(01/13/2016, 09:29 PM)andydude Wrote:(01/13/2016, 01:37 PM)sheldonison Wrote: The two pari-gp programs agree with each other. And they both agree that when you rotate 180 degrees around eta, the function you get is no longer real valued at the real axis!
What exactly do you mean by "rotate"? Do you mean if you start with a base \( b > \eta \) and vary the base towards 1 that the Riemann mapping function turns the already real solution into a complex solution?
Start with \( b=\eta+0.2 \)
Then rotate slowly
\( b=\eta+0.2\cdot\exp(0.25 \pi i) ...\;\; b=\eta+0.2\cdot\exp(0.50 \pi i) ...\;\; \eta+0.2\cdot\exp(0.75 \pi i) ... \;\; \eta+0.2\cdot\exp(1.0 \pi i)=\eta-0.2\approx 1.245 \)
At each step along the way, generate an analytic Kneser type mapping using one of the fixed points in the upper half of the complex plane, and the other fixed point in the lower half of the complex plane. The resulting Kneser mapping is no longer real valued at the real axis. The imaginary offset is much more visible in the slog between the fixed points, which varies by about 2*10^-8. The sexp(z) imaginary component is around 10^-14.
update Better yet, look at the beautiful sequence of complex base tetration in my tetcomplex.gp post#4 and post#15, http://math.eretrandre.org/tetrationforu...hp?tid=729
ok, there is a natural desire to work with b=sqrt(2); So lets load up fatou.gp, and up the precision to 67 decimal digits. The two fixed points are 2 and 4.
Code:
\r fatou.gp
\p 67
sexpinit(sqrt(2)) /* the default limit of 50 iterations also limits precision to 10^-46 */
/* if desired, to get around the 50 iteration count, use loop(log(log(sqrt(2)))+1,70) */
slog(-2) /* these 4 points<2 are near the real axis, but Im(z) isn't exactly zero */
slog(0)
slog(0.5) /* the imaginary jitter is around 10^-48 */
slog(1.9)
slog(2.1) /* these 3 points beween 2 & 4 have Im(z)~=-8.57i, but its not exact */
slog(3) /* the imaginary jitter is around 10^-25 */
slog(3.9)
slog(4.1) /* these 3 points>4 have Im(z)~=-18.2i, but its not exact either */
slog(6) /* the imaginary jitter is around 10^-50 */
slog(8)
- Sheldon