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Solving F(z + 1) = b F(z) in the complex plane

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Abstract

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.

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Authors and Affiliations

  1. Arkansas State University, Arkansas, USA

    William Paulsen & Samuel Cowgill

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  1. William Paulsen
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  2. Samuel Cowgill
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Correspondence to William Paulsen.

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Communicated by: Aihui Zhou

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Paulsen, W., Cowgill, S. Solving F(z + 1) = b F(z) in the complex plane. Adv Comput Math 43, 1261–1282 (2017). https://doi.org/10.1007/s10444-017-9524-1

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  • DOI: https://doi.org/10.1007/s10444-017-9524-1

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