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