File:ValerieDoya.jpg

Photo of Valérie Doya; she had inspired the wide researches of superfunctions and first attempts of their application in laser science in 2008.

Copyright 2008 by Valerie Doya, http://valerie.doya.free.fr/Doya_CV.html The source should be attributed. For the commercial use, communicate the author.

Superfunctions

This picture is used as Fig.5.4 at page 50 of book «Superfunctions» ^[1][2].

For the analysis, the transfer function

\[T(z)=\mathrm{Tania}\big( 1 + \mathrm{ArcTania}(z) \big) \]

needs some specific notation. Here, \( \mathrm{ArcTania}(z) = z+\ln(z)-1 \) is elementary function.

Valierie Doya agreed her name to be used as name of this transfer function \(T\).

Then the Tania function appears as superfunction of Doya function;
ArcTania \(=\) Tania\(^{-1}\) appears as Abelfunction of Doya function.

In vicinity of the real axis, Doya function can be expressed through function LambertW:

\[ \mathrm{Doya}(z)=\mathrm{LambertW}\!\left( z\ \mathrm e^{z+1} \right) \]

However, the implementation of Doya function through the LambertW does not allow to plot the beautiful maps in the complex plane.

Consideration of behavior of functions in the complex plane is essential for uniqueness of superfunctions.

The representation through the LambertW cannot substitute the expression through the Tania function.

References

Keywords

«Doya function», «Superfunction», «Superfunctions», «Tania function», «Transfer function»,