File:Qexpmap.jpg

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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

↑ http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf D.Kouznetsov1, H.Trappmann. Superfunctions and sqrt of Factorial. (2010)
↑ 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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