09/30/2008, 07:58 AM
(This post was last modified: 09/30/2008, 07:59 AM by Kouznetsov.)

Uniqueness of analytic tetration, scheme of the proof.

Notations.

Let be set of complex numbers.

Let be set of real numbers.

Let be set of integer numbers.

Let

Let be base of tetration.

Assume that there exist analytic tetration on base , id est,

(0) is analytic at

(1) for all , the relation holds

(2)

(3) is real increasing function at .

Properties.

From assumption (1) and (2) it follows, that function has singularity at -2, at -3 and so on.

Consider following Assumption:

There exist entire 1-periodic function such that

is also analytic tetration on base .

Then, function is not allowed to take values -2, -3, ..

being evaluated at elements of .

This means that function

is entire function.

(Weak statement which seems to be true)

Function cannot grow faster than linear function at infinity in any direction.

Therefore, it is linear function. Therefore,

Therefore, there exist only one analytic tetration.

Notations.

Let be set of complex numbers.

Let be set of real numbers.

Let be set of integer numbers.

Let

Let be base of tetration.

Assume that there exist analytic tetration on base , id est,

(0) is analytic at

(1) for all , the relation holds

(2)

(3) is real increasing function at .

Properties.

From assumption (1) and (2) it follows, that function has singularity at -2, at -3 and so on.

Consider following Assumption:

There exist entire 1-periodic function such that

is also analytic tetration on base .

Then, function is not allowed to take values -2, -3, ..

being evaluated at elements of .

This means that function

is entire function.

(Weak statement which seems to be true)

Function cannot grow faster than linear function at infinity in any direction.

Therefore, it is linear function. Therefore,

Therefore, there exist only one analytic tetration.