\( \alpha_1(\text{tet}_b(z)) = z + \theta_1(z) \)
\( \alpha_2(\text{tet}_b(z)) = z + \theta_2(z) \)
for b = 3 + i.
This is the main part and I agree on that. No arguing there.
Forget about the Riemann mapping comments.
After some consideration this is not so bad afterall.
How do we call this ?
system of 2 functional equations ?
coupled functional equations ?
But the core question is : how do we solve this :
\( \alpha_1(\text{tet}_b(z)) = z + \theta_1(z) \)
\( \alpha_2(\text{tet}_b(z)) = z + \theta_2(z) \)
for b = 3 + i.
?
The equation seemed familiar to me, so I looked in my very old notebook and ...
I found the related following 2 equivalent equations :
1)
[A_1(f(z)) - A_1(f(z-1))] / [A_2(f(z)) - A_2(f(z-1))] = 1
and
2)
[A_1(f(z)) - A_1(f(z-1))] - [A_2(f(z)) - A_2(f(z-1))] = 0
where the A_i where given functions.
(the [] were written not as brackets but by a difference symbol equivalent to newton's though that does not matter)
Now it turns out I tried to solve 2) by using a (truncated) taylor series for A_1,A_2 and f(z).
In more modern context I guess that means using truncated carleman matrices !
Afterall since equation 2) only has -,+ and composition , carleman matrices seem perfect !
A bit later I wrote the comment : " fibonacci like " ?
Which still makes me wonder today.
Is there a better method than carleman/taylor in the style of fibonacci ?
And of course, the reason I write all this :
Do sheldon and mike use the same method to solve this " coupled functional equation " ?
It appears the equations 1) and 2) are more general than the original equations ...
...
for instance say A_1(f(z)) - A_1(f(z-1)) = z^2 = A_2(f(z)) - A_2(f(z-1)) then we also get :
[A_1(f(z)) - A_1(f(z-1))] / [A_2(f(z)) - A_2(f(z-1))] = 1
and
[A_1(f(z)) - A_1(f(z-1))] - [A_2(f(z)) - A_2(f(z-1))] = 0
But NOT a solution we want.
All a bit confusing ...
But very intresting !
Reminds me a little bit of " the fermat superfunction " too.
2 fixpoints are intresting it seems.
http://math.eretrandre.org/tetrationforu...hp?tid=809
regards
tommy1729
\( \alpha_2(\text{tet}_b(z)) = z + \theta_2(z) \)
for b = 3 + i.
This is the main part and I agree on that. No arguing there.
Forget about the Riemann mapping comments.
After some consideration this is not so bad afterall.
How do we call this ?
system of 2 functional equations ?
coupled functional equations ?
But the core question is : how do we solve this :
\( \alpha_1(\text{tet}_b(z)) = z + \theta_1(z) \)
\( \alpha_2(\text{tet}_b(z)) = z + \theta_2(z) \)
for b = 3 + i.
?
The equation seemed familiar to me, so I looked in my very old notebook and ...
I found the related following 2 equivalent equations :
1)
[A_1(f(z)) - A_1(f(z-1))] / [A_2(f(z)) - A_2(f(z-1))] = 1
and
2)
[A_1(f(z)) - A_1(f(z-1))] - [A_2(f(z)) - A_2(f(z-1))] = 0
where the A_i where given functions.
(the [] were written not as brackets but by a difference symbol equivalent to newton's though that does not matter)
Now it turns out I tried to solve 2) by using a (truncated) taylor series for A_1,A_2 and f(z).
In more modern context I guess that means using truncated carleman matrices !
Afterall since equation 2) only has -,+ and composition , carleman matrices seem perfect !
A bit later I wrote the comment : " fibonacci like " ?
Which still makes me wonder today.
Is there a better method than carleman/taylor in the style of fibonacci ?
And of course, the reason I write all this :
Do sheldon and mike use the same method to solve this " coupled functional equation " ?
It appears the equations 1) and 2) are more general than the original equations ...
...
for instance say A_1(f(z)) - A_1(f(z-1)) = z^2 = A_2(f(z)) - A_2(f(z-1)) then we also get :
[A_1(f(z)) - A_1(f(z-1))] / [A_2(f(z)) - A_2(f(z-1))] = 1
and
[A_1(f(z)) - A_1(f(z-1))] - [A_2(f(z)) - A_2(f(z-1))] = 0
But NOT a solution we want.
All a bit confusing ...
But very intresting !
Reminds me a little bit of " the fermat superfunction " too.
2 fixpoints are intresting it seems.
http://math.eretrandre.org/tetrationforu...hp?tid=809
regards
tommy1729