Difference between revisions of "File:Qexpmap.jpg"

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{{oq|Qexpmap.jpg|Original file ‎(682 × 706 pixels, file size: 152 KB, MIME type: image/jpeg)}}
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[[Complex map]] of function [[square root of exponential]].
  
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  ⚫
[[Complex map]] of function «[[square root of exponential]]», \( \varphi=\sqrt{\exp} \),
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from article «[[Superfunctions and square root of factorial]]»
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<ref name="factorial">
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https://link.springer.com/article/10.3103/S0027134910010029 <br>
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https://mizugadro.mydns.jp/PAPERS/2010superfae.pdf (English)<br>
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https://mizugadro.mydns.jp/PAPERS/2010superfar.pdf (Russian)<br>
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D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12. (Russian version: p.8-14)
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</ref>. In the article, the [[square root of exponential]] (shown in the map) is compared to the [[square root of factorial]].
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  +
Function \( \varphi(z) = \mathrm{tet}(1/2+\mathrm{ate}(z)) = \exp^{1/2}(z)\)
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  +
appears as solution of equation
  +
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\(
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\varphi(\varphi(z))=\exp(z)
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\)
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Existence of this function is shown in 1950 by [[Hellmuth Kneser]]
 
<ref>
  
<ref>
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http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002175851&physid=phys63#navi [[Hellmuth Kneser]]. Reelle analytische Lösungen der Gleichung \( φ(φ(x))=e^x \) und verwandter Funktionalgleichungen. Journal für die reine und angewandte Mathematik / Zeitschriftenband (1950) / Artikel / 56 - 67
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</ref>.
   
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Natural [[Tetration]] \(\mathrm{tet}\) and
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[[Arctetration]] \(\mathrm{ate}\!=\!\mathrm{tet}^{-1}\)
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are implemented in 2009-2010 <ref>
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http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html <br>
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https://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf
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D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
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Preprint: 2009analuxpRepri.pdf
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</ref><ref>
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http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html <br>
  +
https://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf
  +
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
 
</ref>
  
</ref>
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and described in book «[[Superfunctions]]»
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<ref>
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https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas,algorithms,tables,graphics ペーパーバック – 2020/7/28
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</ref><ref>https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas, algorithms, tables, graphics. Publisher: [[Lambert Academic Publishing]].
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</ref> (chapters 14 and 15), similar map appears in the left part of figure 15.5 at page 211 for iterate \(n\!=\!0.5\ \).
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==Misprint==
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  +
There is a misprint in this map:
  +
  +
\( p=-1.6 \)
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  +
appears instead of correct
  +
  +
\( p=-0.6 \)
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==Improved version==
  +
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The improved (and corrected) version of this map is loaded as
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https://mizugadro.mydns.jp/t/index.php/File:QexpMap1263x1243.png
   
 
==References==
  
==References==
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{{ref}}
−
<references/>
  
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{{fer}}
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==Keywords==
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«[[Abelfunction]]»,
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<b>«[[ArcTetration]]»</b>,
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«[[Book]]»,
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«[[BookMap]]»,
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<b>«[[Hellmuth Kneser]]»</b>,
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«[[Iterate]]»,
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«[[Iteration half of exponential]]»,
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«[[Iteration half of factorial]]»,
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«[[Natural tetration]]»,
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<b>«[[Square root of exponential]]»</b>,
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«[[Square root of factorial]]»,
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«[[Superfunction]]»,
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«[[Superfunctions]]»,
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«[[Tetration]]»,
   
 
[[Category:Abelfunction]]
  
[[Category:Abelfunction]]
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[[Category:ArcTetration]]
 
[[Category:Book]]
  
[[Category:Book]]
 
[[Category:BookMap]]
  
[[Category:BookMap]]
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[[Category:Complex map]]
 
[[Category:Hellmuth Kneser]]
  
[[Category:Hellmuth Kneser]]
 
[[Category:Iterate]]
  
[[Category:Iterate]]
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[[Category:Iteration half of exponential]]
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[[Category:Misprint]]
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[[Category:Natural tetration]]
 
[[Category:Superfunction]]
  
[[Category:Superfunction]]
 
[[Category:Tetration]]
  
[[Category:Tetration]]

Latest revision as of 12:58, 14 December 2025


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 \(n\!=\!0.5\ \).

Misprint

There is a misprint in this map:

\( p=-1.6 \)

appears instead of correct

\( p=-0.6 \)

Improved version

The improved (and corrected) version of this map is loaded as

https://mizugadro.mydns.jp/t/index.php/File:QexpMap1263x1243.png

References

  1. ↑ https://link.springer.com/article/10.3103/S0027134910010029
    https://mizugadro.mydns.jp/PAPERS/2010superfae.pdf (English)
    https://mizugadro.mydns.jp/PAPERS/2010superfar.pdf (Russian)
    D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12. (Russian version: p.8-14)
  2. ↑ http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002175851&physid=phys63#navi Hellmuth Kneser. Reelle analytische Lösungen der Gleichung \( φ(φ(x))=e^x \) und verwandter Funktionalgleichungen. Journal für die reine und angewandte Mathematik / Zeitschriftenband (1950) / Artikel / 56 - 67
  3. ↑ http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
    https://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670. Preprint: 2009analuxpRepri.pdf
  4. ↑ http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
    https://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
  5. ↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics ペーパーバック – 2020/7/28
  6. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.

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