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

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.


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