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 =\exp_b(F(z)) )
holds
(2)=1)
(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
=F(z+h(z)) )
is also analytic tetration on base .
Then, function=z+h(z))
is not allowed to take values -2, -3, ..
being evaluated at elements of.
This means that function
=I(z) \frac{z+2}{I(z)+2} =(z+h(z)) \frac{z+2}{z+h(z)+2} )
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
Let
Let
Let
Let
Assume that there exist analytic tetration
(0)
(1) for all
(2)
(3)
Properties.
From assumption (1) and (2) it follows, that function
Consider following Assumption:
There exist entire 1-periodic function
Then, function
being evaluated at elements of
This means that function
is entire function.
(Weak statement which seems to be true)
Function
Therefore, it is linear function. Therefore,
Therefore, there exist only one analytic tetration.