File:Qexpmap.jpg

Complex map of function «square root of exponential», \( \varphi=\sqrt{\exp} \), from article «Superfunctions and square root of factorial» ^[1]. In the article, the square root of exponential (shown in the map) is compared to the square root of factorial.

Function \( \varphi(z) = \mathrm{tet}(1/2+\mathrm{ate}(z)) = \exp^{1/2}(z)\)

appears as solution of equation

\( \varphi(\varphi(z))=\exp(z) \)

Existence of this function is shown in 1950 by Hellmuth Kneser ^[2].

Natural Tetration \(\mathrm{tet}\) and Arctetration \(\mathrm{ate}\!=\!\mathrm{tet}^{-1}\) are implemented in 2009-2010 ^[3][4] and described in book «Superfunctions» ^[5][6] (chapters 14 and 15), similar map appears in the left part of figure 15.5 at page 211 for iterate \(n\!=\!0.5\ \).

References

Keywords

«Abelfunction», «ArcTetration», «Book», «BookMap», «Hellmuth Kneser», «Iterate», «Iteration half of exponential», «Iteration half of factorial», «Natural tetration», «Square root of exponential», «Square root of factorial», «Superfunction», «Superfunctions», «Tetration»,