Entire Function with Logarithmic Asymptotic

Term Entire Function with Logarithmic Asymptotic appears as title of publication at Applied Mathematical Sciences, 2013.

At the first glance, the term «Entire Function with Logarithmic Asymptotic» may appear oxymoronic, since the logarithm has a branch cut, whereas an entire function is single-valued and holomorphic everywhere. The jump of the Logarithm at the cut line cannot be reproduced with an entire function.

Term «Entire Function with Logarithmic Asymptotic» refers to the sectorial asymptotic:
in some sector with vertex at the origin, the function grows in a way, similar to that of Logarithm.
This sector cannot include the negative part of the real axis: there, the Logarithm is not a continuous function.
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 the 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 \(t_1<t<t_2\) , \[ \lim_{r \to +\infty} \Big( \Phi(r\exp(\mathrm i t)) - \log(r\exp(\mathrm i t)) \Big) = 0 \]

Here \(\log\) denotes the principal branch of the logarithm, with branch cut along the negative real axis.
In general, in TORI, the multivalued functions are not considered. In such a way, the meaning in the sense of the principal brach is default.

Superfunctions

The example of an Entire Function with Logarithmic Asymptotic is described also in book "Superfunctions" .

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.

The maps above show function \(f(x\!+\!\mathrm i y) = u + \mathrm i v\ \) with
lines \(u=\rm const \) and
lines \(v=\rm const \)
in the \(x,y\) plane for \(f\!=\!\log\) and for \(f\!=\!\Phi \).

The qualitative geometry of these level lines is consistent with the defining properties of an Entire Function with Logarithmic Asymptotic.
For function \(f\!=\!\Phi \), the parameters in the definition above can be set as follows: \(t_1=-\pi \ \), \(\ t_2=\pi \ \).
In such a way function \(\Phi \) approaches the Logarithm along almost all sectors, except an arbitrary narrow sector along the negative part of the real axis.
To year 2026, up to knowledge of Editor and of ChatGPT, no other function with such a property had been reported.

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

https://en.wikipedia.org/wiki/Entire_function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. ..

https://en.wikipedia.org/wiki/Oxymoron An oxymoron (plurals: oxymorons and oxymora) is a figure of speech that juxtaposes concepts with opposite meanings within a word or in a phrase that is a self-contradiction. ..

https://mathworld.wolfram.com/BranchCut.html A branch cut is a curve (with ends possibly open, closed, or half-open) in the complex plane across which an analytic multivalued function is discontinuous. For convenience, branch cuts are often taken as lines or line segments. Branch cuts (even those consisting of curves) are also known as cut lines (Arfken 1985, p. 397), slits (Kahan 1987), or branch lines.

Keywords

«Branch cut», «ChatGPT», «Cut line», «Entire function», «Entire Function with Logarithmic Asymptotic», «Logarithm», «LambertW.cin», «Oxymoron», «Sectorial asymptotic», «Sutran.cin», «SuZex.cin», «Superfunction», «Superfunctions», «SuTra», «SuZex», «Tania.cin», «Trappmann function»,