Theorem on increment of tetration

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

Theorem on increment of tetration is statement about asymptotic behavior of solution of the Transfer equation with exponential transfer functions.

The matter of the Theorem is coincidence of two curves (thick red and thin black) in Fig.1. These curves represent imaginary part of the growing factor \(K\) and that of the increment \(k\) of the asymptotic solution versus logarithm of the base. From the first glance, the overlapping of the curves may look strange. The Theorem indicates, that it is not an occasional coincidence, but the unavoidable equality.

The therorem 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

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

Proof

Substitution of the asymptotic of solutioun \(F\) into the transfer equation gives

\( K=\beta L = \ln(\beta) \)

Then, \( \exp(k)=\beta L\)

\( k=\ln(\beta\, L) =\) \(\ln(L) + \ln(\beta) =\) \(K + \ln(\beta) \)

While \( \beta >0 \), its logarithm is real.
Hence, the imaginary parts of \(k\) and \(K\) are the same.

(End of proof)

The quantities mentioned are plotted in Fig.1 as functions of \(\beta\)

Applications

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

\(L=L_2=\) 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, Russian version.
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=vmj&paperid=10&option_lang=eng D.Yu.Kuznetsov. “Tetrational as special function”, Vladikavkaz. Mat. Zh., 12:2 (2010), 31–45, English version.

http://link.springer.com/article/10.1007/s10444-017-9524-1 William Paulsen and Samuel 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