01/24/2021, 01:28 AM
(This post was last modified: 01/24/2021, 01:39 AM by sheldonison.)
so, \( \phi(s+1)=e^{\phi(s)+s \), and we're looking for singularities where \( \ln(\ln(\phi(s+1))) \) has a singularity where
\( \ln(\phi(s)+1)=0;\;\;\;\phi+s=0; \) cause then \( \ln(\phi(s)+s) \) has a singularity.
The conjecture is that somewhere nearby the neighborhood of the point where \( \phi(s-1)+s-1=2(n-1)\pi i \), then for a reasonably small value of x, especially as the integer n gets arbitrarily large, we will have \( \phi(s+x)+s+x=0 \), and we will have our singularity. The idea is we are looking for singularities where Re(s) is positive so Re(phi(s)) must be negative to counter it and generate the desired zero value. So that's why the singularities are near odd multiples of pi i for phi(s-1)+s-1. Meanwhile, at the real axis phi is getting quite large, so this is a way to find the places where |phi| is actually relatively small. The conjecture is further that there is a 1:1 correspondence with with these values of s, and the singularities in ln(ln(phi(s+1)) closest to the real axis.
Then here are the first 10 singularities ... notice that as conjectured, the singularities are pretty close to where phi(s-1)+s-1=(2n-1)pi i, just a bit closer to the real axis, and slightly smaller in real magnitude as well. So then these singularities act a lot like the singularities in the base change tetration, or in Peter Walker's slog, that we have discussed before on this forum. These functions are \( C_\infty \) and conjectured to be nowhere analytic since the imaginary part of the value of s where there is a singularity gets arbitrarily small as n gets larger, and since the singularities quickly get arbitrarily close together as n increases as well, so as n increases, \( \text{Tet}_\phi(s+k)=\lim_{n\to\infty}\ln^{[\circ n]}(\phi(s+n)) \) we encounter a wall of singularities.
\( \ln(\phi(s)+1)=0;\;\;\;\phi+s=0; \) cause then \( \ln(\phi(s)+s) \) has a singularity.
The conjecture is that somewhere nearby the neighborhood of the point where \( \phi(s-1)+s-1=2(n-1)\pi i \), then for a reasonably small value of x, especially as the integer n gets arbitrarily large, we will have \( \phi(s+x)+s+x=0 \), and we will have our singularity. The idea is we are looking for singularities where Re(s) is positive so Re(phi(s)) must be negative to counter it and generate the desired zero value. So that's why the singularities are near odd multiples of pi i for phi(s-1)+s-1. Meanwhile, at the real axis phi is getting quite large, so this is a way to find the places where |phi| is actually relatively small. The conjecture is further that there is a 1:1 correspondence with with these values of s, and the singularities in ln(ln(phi(s+1)) closest to the real axis.
Then here are the first 10 singularities ... notice that as conjectured, the singularities are pretty close to where phi(s-1)+s-1=(2n-1)pi i, just a bit closer to the real axis, and slightly smaller in real magnitude as well. So then these singularities act a lot like the singularities in the base change tetration, or in Peter Walker's slog, that we have discussed before on this forum. These functions are \( C_\infty \) and conjectured to be nowhere analytic since the imaginary part of the value of s where there is a singularity gets arbitrarily small as n gets larger, and since the singularities quickly get arbitrarily close together as n increases as well, so as n increases, \( \text{Tet}_\phi(s+k)=\lim_{n\to\infty}\ln^{[\circ n]}(\phi(s+n)) \) we encounter a wall of singularities.
Code:
s1,s2 = 3.08188+1.191682*I, 2.69716+1.019425*I; phi((s1-1)+s1-1= 1pi*i; phi(s2)+s2=0 |s1-s2|=0.421523
s1,s2 = 3.12347+0.563422*I, 3.09274+0.531645*I; phi((s1-1)+s1-1= 3pi*i; phi(s2)+s2=0 |s1-s2|=0.044203
s1,s2 = 3.21853+0.435711*I, 3.20564+0.419751*I; phi((s1-1)+s1-1= 5pi*i; phi(s2)+s2=0 |s1-s2|=0.020517
s1,s2 = 3.27739+0.374627*I, 3.26990+0.364341*I; phi((s1-1)+s1-1= 7pi*i; phi(s2)+s2=0 |s1-s2|=0.012728
s1,s2 = 3.31871+0.337247*I, 3.31366+0.329806*I; phi((s1-1)+s1-1= 9pi*i; phi(s2)+s2=0 |s1-s2|=0.008994
s1,s2 = 3.35001+0.311396*I, 3.34630+0.305640*I; phi((s1-1)+s1-1=11pi*i; phi(s2)+s2=0 |s1-s2|=0.006847
s1,s2 = 3.37493+0.292151*I, 3.37206+0.287496*I; phi((s1-1)+s1-1=13pi*i; phi(s2)+s2=0 |s1-s2|=0.005470
s1,s2 = 3.39548+0.277098*I, 3.39317+0.273215*I; phi((s1-1)+s1-1=15pi*i; phi(s2)+s2=0 |s1-s2|=0.004521
s1,s2 = 3.41286+0.264900*I, 3.41094+0.261585*I; phi((s1-1)+s1-1=17pi*i; phi(s2)+s2=0 |s1-s2|=0.003831
s1,s2 = 3.42786+0.254750*I, 3.42623+0.251868*I; phi((s1-1)+s1-1=19pi*i; phi(s2)+s2=0 |s1-s2|=0.003309
- Sheldon