Adv Comput Math https://doi.org/10.1007/s10444-018-9615-7 William Paulsen Received: 9 January 2018 / Accepted: 22 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract In this paper we will consider the tetration, defined by the equation F(z + F(z) 1) = b 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 1/e real and b> 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. Keywords Tetration · Abel’s functional equation · Iteration Mathematics Subject Classification (2010) 26A18 · 30D05 · 39B12 1 Background Much research has already been done on the tetration function, whose name comes from tetra- (four) and iteration. Addition by a positive integer n can be thought of as repeated incrementing, multiplication by n is done by repeated addition, and Communicated by: Aihui Zhou William Paulsen firstname.lastname@example.org Arkansas State University, Jonesboro, AR, USA W. Paulsen exponentiation by n is repeated multiplication. So a
Advances in Computational Mathematics – Springer Journals
Published: Jun 2, 2018
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