10/26/2016, 08:01 AM
(This post was last modified: 10/26/2016, 08:48 AM by sheldonison.)
(10/24/2016, 11:50 PM)Gottfried Wrote: By some other idea I investigated the behave near above the real axis, and specifically I looked at the point \( \small z_0= \text{Re} (L) + \varepsilon I \) and found, that the required height \( h \) to iterate from \( z_0 \) to\( L \) is nearly \( \small 2 \pi I \) , so
\( \small \text{sexp}(2 \pi I+ \text{slog}(z_0 ) )\approx L \qquad \qquad \text{ using }z_0 \text{ with } \varepsilon=1e-6,1e-12,1e-24,... \)
That this is very near suggests, that here equality is intended....
Gottfried,
You're using a real base>exp(1/e). Either base(2) or base(e), so you don't really need the complex base support in tetcomplex. Kneser.gp would work just fine, and as you noted, tetcomplex won't work with the latest versions of pari-gp.
For any real valued sexp base as imag(z) increases, sexp(z) will go to the fixed point, since the Kneser tetration function is a 1-cyclic mapping of S(z) where S(z) is the Schröder solution superfunction from the complex fixed point. By definition, \( S(z+1)=\exp_b(S(z))\;\; \). In the upper half of the complex plane, we use \( S_U(z) \) and in the lower half of the complex plane we use \( S_L(z) \) generated from the complex conjugate fixed point.
\( \text{If}\; \Im(z)>0 \;\;\text{sexp}(z)=\text{S}_U(z+\theta_U(z)) \)
\( \text{If} \; \Im(z)<0 \;\;\text{sexp}(z)=\text{S}_L(z+\theta_L(z)) \)
The solution in the upper half of the complex plane approaches \( S_U(z+k) \) as \( \Im(z) \) increases. This is because the periodic terms in the 1-cyclic \( \theta_U(z) \) mapping quickly decay to zero as imag(z) increases leaving only the constant term of k. Try plugging in z=2*I into this equation for theta, to see how quickly the periodic terms in \( \theta_U(z) \) decay.
\( \theta_U(z) = k + \sum_{n=1}^{\infty} \; a_n \exp(2n\pi i z) \)
The \( S_U(z)\;S_L(z) \) functions are also periodic with a different period that decays more slowly. For base2 the Su(z) Schröder solution superfunction has a period of 5.584 + 1.0542*I, and Su(z) decays to the fixed point as \( \Im(z) \) increases, but the periodic terms in Su(z) decay much more slowly than the \( \theta_U(z) \) function.
At the real axis \( \theta_U(z) \) has a really complicated singularity at integer values of z, since the Schröder superfunction solution never takes on the values of 0 anywhere in the complex plane, and yet sexp(-1)=0. So \( \theta_U(z) \) is only used in the upper half of the complex plane; and its complex conjugate is used in the lower half of the complex plane.
So sexp_2(2*Pi*I) = 0.827783486030388 + 1.56642681578434*I; and the fixed point is 0.824678546142074 + 1.56743212384965*I.
abs(sexp_2(2*Pi*I)-L)=0.00326
abs(sexp_2(10*I)-L)=0.0000572
abs(sexp_2(20*I)-L)=1.101E-9
abs(sexp_2(30*I)-L)=2.105E-14
I hope this helps.
- Sheldon
