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

@article{Paulsen2017SolvingF, title={Solving F(z + 1) = bF(z) in the complex plane}, author={William H. Paulsen and Samuel Cowgill}, journal={Advances in Computational Mathematics}, year={2017}, volume={43}, pages={1261-1282} }

The generalized tetration, defined by the equation F(z+1) = bF(z) in the complex plane with F(0) = 1, is considered for any b > e1/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…

## 3 Citations

Tetration for complex bases

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- 2019

The tetration, defined by the equation F(z + 1) = bF(z) in the complex plane with F(0) =1, for the case where b is complex, is considered.

Constructing a hyperoperation sequence-pisa hyperoperations

- Mathematics
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In the context of other hyperoperation sequences, a new sequence of operations is constructed. A review of its properties reveals dependences between pairs of numbers, and so the sibling numbers are…

Extension of tetration to real and complex heights

- Mathematics
- 2021

The continuous tetrational function r = τ(r, x), the unique solution of equation τ(r, x) = r and its differential equation τ (r, x) = qτ(r, x)τ (r, x − 1), is given explicitly as r = exp r [{x}]q,…

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