01/19/2016, 01:24 PM
The Böttcher function does not really help much.
It is just the double exponential of the Abel function.
Or the exponential of A generalized Koenigs function.
Consider j(x)
Abel_1(j(x)) = Abel_1(x) + 1.
A general Abel_2 is of the form
Abel_2(x) = Abel_1(x) + invar(x).
Where invar is the invariant :
Invar(j(x)) = Invar(x).
Invar relates trivially to the theta function used for going from
Super_1 to Super_2.
The general Böttcher is then basically a double exponential of this general Abel.
So basically we just solved Abel Or Schröeder equation , and reached nothing more than the usual Koenigs and the understanding of the theta / invar.
Anything beyond is conjectural in nature , and does not occur in the works of Böttcher.
I assume your computation is just like this and thus not so helpfull ( in the context of this thread ).
This is the reason i did not mention Böttcher before.
His work seems not to help with matters like radius of convergeance , area of agreement with functional equation , agreement on fixpoints etc etc.
Sorry Im skeptical.
As often imho An example of a mathematician named after his least contribution ;
His other work is imho more intresting.
( other examples are the forgotten work of Cantor , theorem named after people who did not prove it or conjectured it first etc )
We wrote alot about the theta wave and the invar.
It seems logical to assume that a crucial invariant for base sqrt 2 should be of the form ( for validity in 2 < x < 4 ) :
Sum Q(g^[k](x))
Where K runs over the integers and Q(2) = Q(4) = 0.
Regards
Tommy1729
It is just the double exponential of the Abel function.
Or the exponential of A generalized Koenigs function.
Consider j(x)
Abel_1(j(x)) = Abel_1(x) + 1.
A general Abel_2 is of the form
Abel_2(x) = Abel_1(x) + invar(x).
Where invar is the invariant :
Invar(j(x)) = Invar(x).
Invar relates trivially to the theta function used for going from
Super_1 to Super_2.
The general Böttcher is then basically a double exponential of this general Abel.
So basically we just solved Abel Or Schröeder equation , and reached nothing more than the usual Koenigs and the understanding of the theta / invar.
Anything beyond is conjectural in nature , and does not occur in the works of Böttcher.
I assume your computation is just like this and thus not so helpfull ( in the context of this thread ).
This is the reason i did not mention Böttcher before.
His work seems not to help with matters like radius of convergeance , area of agreement with functional equation , agreement on fixpoints etc etc.
Sorry Im skeptical.
As often imho An example of a mathematician named after his least contribution ;
His other work is imho more intresting.
( other examples are the forgotten work of Cantor , theorem named after people who did not prove it or conjectured it first etc )
We wrote alot about the theta wave and the invar.
It seems logical to assume that a crucial invariant for base sqrt 2 should be of the form ( for validity in 2 < x < 4 ) :
Sum Q(g^[k](x))
Where K runs over the integers and Q(2) = Q(4) = 0.
Regards
Tommy1729