Tetration for complex bases

Tetration for complex bases 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 wpaulsen@astate.edu Arkansas State University, Jonesboro, AR, USA W. Paulsen exponentiation by n is repeated multiplication. So a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Computational Mathematics Springer Journals

Tetration for complex bases

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Publisher
Springer US
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics; Mathematical and Computational Biology; Computational Science and Engineering; Visualization
ISSN
1019-7168
eISSN
1572-9044
D.O.I.
10.1007/s10444-018-9615-7
Publisher site
See Article on Publisher Site

Abstract

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 wpaulsen@astate.edu Arkansas State University, Jonesboro, AR, USA W. Paulsen exponentiation by n is repeated multiplication. So a

Journal

Advances in Computational MathematicsSpringer Journals

Published: Jun 2, 2018

References

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