View source for Entire Function with Logarithmic Asymptotic

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Term [[Entire Function with Logarithmic Asymptotic]] appears as title of publication  
<ref name="hikari">
https://www.m-hikari.com/ams/ams-2013/ams-129-132-2013/kouznetsovAMS129-132-2013.pdf <br>
https://mizugadro.mydns.jp/PAPERS/2013hikari.pdf
D.Kouznetsov. [[Entire Function with Logarithmic Asymptotic]]. [[Applied Mathematical Sciences]], Vol. 7, 2013, no. 131, 6527 - 6541
</ref> at [[Applied Mathematical Sciences]], 2013.

Term [[Entire Function with Logarithmic Asymptotic]] can make an impression of an [[oxymoron]]:<br>
function logarithm has cut line, that cannot be reproduced with an [[entire function]]. 

Term [[Entire Function with Logarithmic Asymptotic]]
refers to the [[sectorial asymptotic]]: <br>
in some sector with vertex at the origin,
the function grows in a way, similar to that of [[Logarithm]].<br>
This sector cannot include the negative part of the real axis: there, the Logarithm is not a continuous function.<br>
However the sector may cover almost all the complex plane, excluding an arbitrary narrow sector along the negative part of the real axis.

==Definition==

[[Entire Function with Logarithmic Asymptotic]] 
is function \(\Phi\) of complex argument such that

(1)  \(\Phi\) is holomorphic in the whole complex plane
 
(2) in some sector of the complex plane \(\Phi\) approach function [[Logarithm]],
id est, there exist real values \(t_1\) and \(t_2\)
such that  \(-\pi<t_1<t_2<\pi\)
and for any fixed real \(t\) with \(t_1<t<t_2\) ,
\[
\lim_{r \to +\infty} \Big(
\Phi(r\exp(\mathrm i t)) - \log(r\exp(\mathrm i t)) 
\Big) = 0
\]

==[[Superfunctions]]==
The example <ref name="hikari"/>
of an 
[[Entire Function with Logarithmic Asymptotic]]
is described also in book "[[Superfunctions]]" <ref><small>
https://www.amazon.com/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862
</small><br>
https://mizugadro.mydns.jp/BOOK/468.pdf
D.Kouznetsov. [[Superfunctions]].  [[Lambert Academic Publishing]], 2020.
</ref><ref>
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
https://mizugadro.mydns.jp/BOOK/202.pdf
Д.Кузнецов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014 (In Russian)
</ref>.

The [[Entire Function with Logarithmic Asymptotic]] is expressed through the [[Superfunction]]
[[SuTra]] of the elementary [[Trappmann function]] \(\mathrm{tra} = z \mapsto z+\exp(z) \):

\[
\Phi(z) = - \mathrm{SuTra} (-z)
\]
The [[complex map]] of [[Logarithm]] and that of function \(\Phi\) above are shown in the pictures below.<br>
{{pic|SuTraMapLT.png|400px}}
{{pic|SuTraMapN.png|400px}}<br>
The maps above show function \(f(x\!+\!\mathrm i y) = u + \mathrm i v\ \) with <br>
lines \(u=\rm const \) and<br>
lines \(v=\rm const \) <br>
in the \(x,y\) plane for \(f\!=\!\log\) and for \(f\!=\!\Phi \).

The qualitative geometry of these level lines confirms the main claim about the [[Entire Function with Logarithmic Asymptotic]].

==Notes by [[ChatGPT]]==

The example \(\Phi(z)=-\mathrm{SuTra}(-z)\) described in this article does not appear to be isolated.

A natural way to obtain further candidates for an [[Entire Function with Logarithmic Asymptotic]] is to replace the [[Trappmann function]]
\[
\mathrm{tra}(z)=z+\exp(z)
\]
by a more general [[transfer function]] of the form
\[
T_{a,b}(z)=z+a\ \exp(bz),
\qquad a\neq 0,\ b\neq 0,
\]
and to construct a corresponding [[superfunction]] \(F_{a,b}\) satisfying
\[
T_{a,b}(F_{a,b}(z))=F_{a,b}(z+1),
\]
with an asymptotic normalization
\[
F_{a,b}(z)\sim -\tfrac{1}{b}\log(-z)+C
\quad\text{as }\Re z\to -\infty.
\]

If such a [[superfunction]] \(F_{a,b}\) exists and is entire, then the function
\[
\Phi_{a,b}(z)=-F_{a,b}(-z)
\]
formally satisfies a logarithmic asymptotic
\[
\Phi_{a,b}(z)\sim \tfrac{1}{b}\log z - C
\]
in any sector avoiding the negative real axis.

In this way, logarithmic behavior arises not from analytic continuation of the logarithm itself, but as an asymptotic inverse of exponential dynamics under iteration. This mechanism explains how an entire function may mimic the logarithm in wide sectors of the complex plane while remaining single-valued and holomorphic everywhere.

At present, no examples of [[Entire Function with Logarithmic Asymptotic]] are known that do not arise, directly or indirectly, from such [[superfunction]] constructions. It is also not known whether all entire functions with logarithmic asymptotic must be obtainable in this way. These questions remain open.

==Acknowledgement==
This article is created following the advice by [[ChatGPT]].

Some stylistic suggestions by [[ChatGPT]] are taken into account.

The section above is generated by [[ChatGPT]] after the request from Editor.

==References==
{{ref}}

{{fer}}

==Keywords==
«[[ChatGPT]]»,
«[[Entire Function with Logarithmic Asymptotic]]»,
«[[Logarithm]]»,
«[[LambertW.cin]]», 
«[[Tania.cin]]»,
«[[Sutran.cin]]»,
«[[SuZex.cin]]»,
«[[Superfunction]]»,
«[[Superfunctions]]»,
«[[SuTra]]»,
«[[SuZex]]»,
«[[Trappmann function]]»,

[[Category:C++]]
[[Category:ChatGPT]]
[[Category:Complex map]]
[[Category:Entire Function with Logarithmic Asymptotic]]
[[Category:Logarithm]]
[[Category:Superfunction]]
[[Category:Superfunctions]]
[[Category:Trappmann function]]