User talk:Dmitrii Kouznetsov/Analytic Tetration: Difference between revisions

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imported>Dmitrii Kouznetsov
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'''Proof''' see in Reinhold Remmert, "Funktionnentheorie", Springer, 1995. As reference is given: H. Wielandt 1939. (Mein Gott, so old reference!)
'''Proof''' see in Reinhold Remmert, "Funktionnentheorie", Springer, 1995. As reference is given: H. Wielandt 1939. (Mein Gott, so old reference!)


===Theorem T2===
===Theorem T2 (about exponential)===
'''Let''' <math>~E~</math> be solution of  
'''Let''' <math>~E~</math> be solution of  
<math>~ E(z+1)=b E(x)~</math>,  
<math>~ E(z+1)=b E(x)~</math>,  

Revision as of 06:33, 29 September 2008

Henryk Trappmann 's theorems

This is approach to the Second part of the Theorem 0, which is still absent in the main text.

Copypast from http://math.eretrandre.org/tetrationforum/showthread.php?tid=165&pid=2458#pid2458

Theorem T1. (about Gamma funciton)

Let be holomorphic on the right half plane let for all such that .
Let .
Let be bounded on the strip .
Then is the gamma function.

Proof see in Reinhold Remmert, "Funktionnentheorie", Springer, 1995. As reference is given: H. Wielandt 1939. (Mein Gott, so old reference!)

Theorem T2 (about exponential)

Let be solution of , , bounded in the strip .

Then is exponential on base , id est, .

Proof. We know that every other solution must be of the form where is a 1-periodic holomorphic function. This can roughly be seen by showing periodicity of .

,

where is also a 1-periodic function,

While each of and is bounded on , must be bounded too.

Theorem T3

Let .
Let Let Let

Then

Discussion. Id est, is Fibbonachi function.

Theorem T4

Let .
Let each of and satisfies conditions

for
is holomorphic function, bounded in the strip .

Then

Discussion. Such is unique tetration on the base .