User talk:Dmitrii Kouznetsov/Analytic Tetration

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 function)

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F~} be holomorphic on the right half plane let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F(z+1)=zF(z)~} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~z~} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\Re(z)>0~} .
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F(1)=1~} .
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} be bounded on the strip Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~1 \le \Re(z)<2 ~} .
Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F~} is the gamma function.

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

Consider function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~v=F-\Gamma~} on the right half plane, it also satisfies equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~v(z+1)~=~z~v(z)~} Hence, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~v~} has a meromorphic continuation to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\mathbb{C}~} ; and the poles are allowed only at non–positive integer values of the argument.

While Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~v(1)=0~} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\lim_{z\rightarrow 0} z~v(z)=0~} , hence, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~v~} has a holomorphic continuation to 0 and also to each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~-n~} , $~n\in \mathbb {N}$ by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~v(z+1)=z~v(z)~} .

In the range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ 1\le \Re(z) <2 ~} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~v(z)~ } is pounded. It is because function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ \Gamma ~ } is bounded there.

Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~v(z)~} is also restricted on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\mathbb{S}~} , because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~v(z)!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~v(1-z)!} have the same values on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\mathbb{S}~} . Now Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~q(z+1)=-q(z)~} , hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~q~} is bounded on whole Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\mathbb{C}~} , and by the Liouville Theorem, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~q(z)=q(1)=0} . Hence, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~v=0~} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F=\Gamma~} .

(end of proof)

Theorem T2 (about exponential)

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~E~} be solution of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ E(z+1)=b E(x)~} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ E(1)=b ~} , bounded in the strip Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ 0\le \Re(z)<1 ~} .

Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~E~} is exponential on base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~b~} , id est, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~E=\exp_b~} .

Proof. We know that every other solution must be of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~g(z)=f(z+p(z))~ } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ p~} is a 1-periodic holomorphic function. This can roughly be seen by showing periodicity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~h(z)=f^{-1} (g(z))-z~ } .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ f(z+p(z))=b^{z+p(z)}=b^{p(z)}b^z=b^p(z)f(z)~=q(z)f(z) ~} ,

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ q(z)=b^p(z) ~} is also a 1-periodic function,

While each of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f~} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~g~} is bounded on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\mathbb{S}~ } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~q~} must be bounded too.

Theorem T3 (about Fibbonachi)

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\phi=\frac{1+\sqrt{5}}{2}~} .
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F(z+1)=F(z)+F(z-1)~} Let $~F(0)=1~$ Let $~F(1)=1~$

Then $~F(z)={\frac {\phi ^{z}-(1-\phi )^{z}}{\sqrt {5}}}$

Discussion. Id est, $~F~$ is Fibbonachi function.

Theorem T4 (about tetration)

First intent to formulate

Let $~b>\exp(1/\mathrm {e} )~$.
Let each of $~F_{1}~$ and $~F_{2}~$ satisfies conditions

$~\exp _{b}(F(z))=F(z+1)~$ for $\Re (z)>-2$

$~F(z)~$ is holomorphic function, bounded in the strip $~|\Re (z)|\leq 1~$ .

Then $~F_{1}=F_{2}~$

Second intent to formulate

(0) Let $~b>\exp(1/\mathrm {e} )~$.

(1) Let each of $~F_{1}~$ and $~F_{2}~$ is holomorphic function on $~\mathbb {C} ^{\prime }=\mathbb {C} \backslash (-\infty ,-2]$, satisfying conditions

(2) $F(0)=1$

(3) $~\exp _{b}(F(z))=F(z+1)~$ for $\Re (z)>-2$

(4) $~F~$ is bounded on $~\mathbb {S} =~\{x+\mathrm {i} y|-2<x\leq 1,y\in \mathbb {R} \}$

Then $~F_{1}=F_{2}~$

Proof of Theorem T4

Lemma 1

(0) Let $~b>\exp(1/\mathrm {e} )~$.

(1) Let $~f~$ be holomorphic function on $~\mathbb {C} ^{\prime }=\mathbb {C} \backslash (-\infty ,-2]~$, such that

(2) $f(0)=1$

(3) $~\exp _{b}(f(z))=f(z+1)~$ for $\Re (z)>-2~$

(4) $~f~$ is bounded on $~\mathbb {S} =~\{x+\mathrm {i} y|-2<x\leq 1,y\in \mathbb {R} \}~$

Let $~\mathbb {D} =~\{x+\mathrm {i} y|-2<x,y\in \mathbb {R} \}~$

Then $~f(\mathbb {D} )=\mathbb {C} ~$

Proof of theorem T4

Henryk, I cannot copypast your proof here: I do not see, where do you use condition

$~b>\exp(1/\mathrm {e} )$

?

From Lemma 1, the ...

Discussion

Such $~F~$ is unique tetration on the base $~b~$.