Tetration for complex bases

Abstract

In this paper we will consider 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. A previous paper determined conditions for a unique solution the case where b is real and b > e1/e. In this paper we extend these results to find conditions which determine a unique solution for complex bases. We also develop iteration methods for numerically approximating the function F(z), both for bases inside and outside the Shell-Thron region.

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Correspondence to William Paulsen.

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Communicated by: Aihui Zhou

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Paulsen, W. Tetration for complex bases. Adv Comput Math 45, 243–267 (2019). https://doi.org/10.1007/s10444-018-9615-7

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Keywords

  • Tetration
  • Abel’s functional equation
  • Iteration

Mathematics Subject Classification (2010)

  • 26A18
  • 30D05
  • 39B12