f(x) = x has 2 solutions : 1 and 2.
The superfunction F(x) is entire and has the fixpoint (of f(x)) 1 @ complex infinity.
Also the other fixpoint 2 occurs for F(x) at - oo.
so everything seems to work out nicely.
But it might be deception !
F(x) = 2
<=> 2^(2^x) +1 = 2
<=> 2^(2^x) = 1
This has MANY solutions.
And that fact seems to make life hard.
We cannot blame singularities now since F(x) is ENTIRE !
Does this imply that F(x) is pseudoperiodic or something ? Or does the functional equation fail ? Both seem to weird to be true.
Now f(x) has 2 fixpoints. So maybe we need 2 superfunctions ?
One seems entire , but to what fixpoint does it belong ?
How does the other superfunction behave ?
What about those methods where we use 2 fixpoints such as the analytic sickel between two fixpoints based on fatou ?
This seems to be as puzzling as tetration itself , hence like I said this is imho " deception ". It is more complicated then it looks.
Seems having the entire property does not solve all issues !
Keep in mind that an answer like " oh thats just because of the log branches " is not a " real " answer.
I was aware of this for a long time but I was waiting for a response to my first post. Since it did not come I felt the need to explain more.
Maybe you agree on the opinion that this is a serious important topic now.
regards
tommy1729
The superfunction F(x) is entire and has the fixpoint (of f(x)) 1 @ complex infinity.
Also the other fixpoint 2 occurs for F(x) at - oo.
so everything seems to work out nicely.
But it might be deception !
F(x) = 2
<=> 2^(2^x) +1 = 2
<=> 2^(2^x) = 1
This has MANY solutions.
And that fact seems to make life hard.
We cannot blame singularities now since F(x) is ENTIRE !
Does this imply that F(x) is pseudoperiodic or something ? Or does the functional equation fail ? Both seem to weird to be true.
Now f(x) has 2 fixpoints. So maybe we need 2 superfunctions ?
One seems entire , but to what fixpoint does it belong ?
How does the other superfunction behave ?
What about those methods where we use 2 fixpoints such as the analytic sickel between two fixpoints based on fatou ?
This seems to be as puzzling as tetration itself , hence like I said this is imho " deception ". It is more complicated then it looks.
Seems having the entire property does not solve all issues !
Keep in mind that an answer like " oh thats just because of the log branches " is not a " real " answer.
I was aware of this for a long time but I was waiting for a response to my first post. Since it did not come I felt the need to explain more.
Maybe you agree on the opinion that this is a serious important topic now.
regards
tommy1729