Difference between revisions of "File:Qexpmap.jpg"

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[[Complex map]] of function [[square root of exponential]].
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[[Complex map]] of function [[square root of exponential]], \( \varphi=\sqrt{\exp} \)
<ref>
+
<ref name="factorial">
  +
https://link.springer.com/article/10.3103/S0027134910010029 <br>
http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf
 
  +
https://mizugadro.mydns.jp/PAPERS/2010superfae.pdf (English)<br>
D.Kouznetsov1, H.Trappmann.
 
  +
https://mizugadro.mydns.jp/PAPERS/2010superfar.pdf (Russian)<br>
Superfunctions and sqrt of Factorial. (2010)
 
  +
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)
 
</ref>
 
</ref>
   
  +
Function \( \varphi(z) = \mathrm{tet}(1/2+\mathrm{ate}(z)) = \exp^{1/2}(z)\)
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  +
appears as solution of equation
  +
  +
\(
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\varphi(\varphi(z))=\exp(z)
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\)
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Existence of this function is shown in 1950 by [[Hellmuth Kneser]]
  +
<ref>
  +
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>
 
==References==
 
==References==
 
<references/>
 
<references/>
   
 
[[Category:Abelfunction]]
 
[[Category:Abelfunction]]
  +
[[Category:ArcTetration]]
 
[[Category:Book]]
 
[[Category:Book]]
 
[[Category:BookMap]]
 
[[Category:BookMap]]

Latest revision as of 02:19, 27 June 2020

Complex map of function square root of exponential, \( \varphi=\sqrt{\exp} \) [1]

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]

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

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