Theorem on increment of tetration

Theorem on increment of tetration is statement about asymptotic behavior of solution of the Transfer equation with exponential transfer functions. It applies not only to tetration, but also to other superexponentials.

Statement

Let \(F\) be solution of equation

\(F(z\!+\!1)=\exp\big(\beta F(z)\big)\)

for some \(\beta>0\).

Let \(L\) be the fixed point, id est, \(\exp(\beta L)=L\)

Let \(F(z)=L+\varepsilon+O(\varepsilon^2) \)

where \(\varepsilon = \exp(kz) \) for some increment \(k\).

Let \(~ K\!=\!\exp(k)\)

Then

\( \Im(K) = \Im(k) \)

Applications

Fig.1. Asymptoric parameters of Tetration versus \(\beta\)

Fig.1 shows the asymptotic parameters of tetration to base \(\ln(\beta)\):

Real and imaginary parts of the fixed points

\(L=L_1=\) Filog\((\beta)\)

and

\(L=L_1=\) Filog\((\beta^*)^*\)

Real and imaginary parts of the asymptotic growing factor

\(K= \beta L\)

Real and imaginary parts of the asymptotic increment

\(k=\ln(K) \)

For \(\beta < 1/\mathrm e\), the two fixed points are shown; and the two values of the corresponding growing factor and two values of the corresponding increment are drown.

For real positive \(\beta\),

The imaginary parts of \(K\) and \(k\) coincide.

References

http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002175851&physid=phys63#navi Hellmuth Kneser. Reelle analytische Lösungen der Gleichung \(\varphi(\varphi(x)=e^x\) und verwandter Funktionalgleichungen. Journal für die reine und angewandte Mathematik / Zeitschriftenband (1950) / Artikel / 56 - 67

http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
http://www.ils.uec.ac.jp/~dima/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.

http://www.vmj.ru/articles/2010_2_4.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45, In Russian. English version:

http://link.springer.com/article/10.1007/s10444-017-9524-1 William Paulsen and amuel Cowgill. Solving F(z+1)=bF(z) in the complex plane. Advances in Computational Mathematics, 2017 March 7, p. 1–22

https://www.morebooks.de/store/gb/book/superfunctions/isbn/978-620-2-67286-3 Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.

Keywords

Asymptotic analysis, Filog, Fixed point, Growing factor, Increment, Kneser Expansion, Superfunction, Tetration