Entire Function with Logarithmic Asymptotic

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

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

In such a way, term Entire Function with Logarithmic Asymptotic refers to the sectorial asymptotic.

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 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 \(\mathrm{tra} = z \mapsto z+\exp(z) \):

\[ \Phi(z) = - \mathrm{SuTra} (-z) \]

References

Keywords

Entire Function with Logarithmic Asymptotic