Solving \(F(z+1)=b^{F(z)}\) in the complex plane. (English) Zbl 1387.30030

Paulsen, William; Cowgill, Samuel

Adv. Comput. Math. 43, No. 6, 1261-1282 (2017).

Summary: The generalized tetration, defined by the equation \(F(z+1)=b^{F(z)}\) in the complex plane with \(F(0)=1\), is considered for any \(b>e^{1/e}\). By comparing other solutions to Kneser’s solution, natural conditions are found which force Kneser’s solution to be the unique solution to the equation. This answers a conjecture posed by Trappmann and Kouznetsov. Also, a new iteration method is developed which numerically approximates the function \(F(z)\) with an error of less than \(10^{-50}\) for most bases \(b\), using only 180 nodes, with each iteration gaining one or two places of accuracy. This method can be applied to other problems involving the Abel equation.

MSC:

30D05	Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
39B12	Iteration theory, iterative and composite equations

Keywords:

tetration; Abel’s functional equation; iteration

References:

[1]	Abel, NH, Untersuchung der functionen zweier unabhängig veränderlichen größen x und y, wie f(x, y), welche die eigenschaft haben, J. Reine Angew. Math., 1, 11-15, (1826) · ERAM 001.0003cj
[2]	Ackermann, W, Zum hilbertschen aufbau der reellen zahlen, Math. Ann., 99, 118-133, (1928) · JFM 54.0056.06
[3]	Atkinson, K., Han, W.: Elementary numerical analysis, 3rd edn. Wiley, New York (2004)
[4]	Jabotinsky, E, Analytic iteration, Trans. Amer. Math. Soc., 108, 457-477, (1963) · Zbl 0113.28303
[5]	Kneser, H, Reelle analytishe Lösungen der gleichung φ(φ(x))=ex und verwandter funktionalgleichungen, J. Reine Angew. Math., 187, 56-67, (1950)
[6]	Koenigs, G, Recherches sur LES intégrales de certaines équations fonctionelles, Ann. Scientifiques l’École Norm. Supérieure, 1, 3-41, (1884) · JFM 16.0376.01
[7]	Kouznetsov, D, Solution of \(F(z+1) = \exp (F(z))\)F(z+1)= exp(F(z)) in the complex z-plane, Math. Comput., 78, 1647-1670, (2009) · Zbl 1202.30002
[8]	Kouznetsov, D, Tetration as special function, Vladikavkaz Math. J., 12, 31-45, (2010) · Zbl 1211.30042
[9]	Kouznetsov, D, Evaluation of holomorphic ackermanns, Appl. Comput. Math., 3, 307-314, (2014)
[10]	Kouznetsov, D; Trappmann, H, Portrait of the four regular super-exponentials to base sqrt(2), Math. Comput., 79, 1727-1756, (2010) · Zbl 1195.30007
[11]	Kuczma, M., Choczewski, B., Ger, R.: Iterative functional equations. Cambridge University Press, Cambridge (1990) · Zbl 0703.39005
[12]	Paulsen, W, Finding the natural solution to \(f(f(x)) = \exp (x)\)f(f(x))= exp(x), Korean J. Math., 24, 81-106, (2016)
[13]	Schröder, E, Über iterierte funktionen, Math. Ann., 2, 296-322, (1871)
[14]	Trappmann, H; Kouznetsov, D, Uniqueness of holomorphic Abel functions at a complex fixed point pair, Aequationes Math., 81, 65-76, (2011) · Zbl 1211.30043
[15]	Walker, P, The exponential of iteration of ex−1, Proc. Am. Math. Soc., 110, 611-620, (1990) · Zbl 0709.39007
[16]	Walker, P, Infinitely differentiable generalized logarithmic and exponential functions, Math Comput., 57, 723-733, (1990) · Zbl 0742.39005
[17]	Walker, P, On the solutions of an abelian functional equation, J. Math. Anal. Appl., 155, 93-110, (1991) · Zbl 0716.39006