Difference between revisions of "File:ShellThronRegionPaulsen2.png"

[[Shell-Thron region]] in the complex plane (inside the loop).

Figure 1 from publication by [[William Harold Paulsen]], 2019

<ref>

https://www.researchgate.net/publication/325532999_Tetration_for_complex_bases <br>

https://link.springer.com/article/10.1007/s10444-018-9615-7

[[William Harold Paulsen]]. Tetration for complex bases.

Advances in Computational Mathematics, volume 45, pages 243–267(2019)

Abstract

In this paper we will consider the tetration, defined by the equation \( F(z+1)= b^F(z)\) 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 \(b>e^{1/e}\).

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. ..

We say that the base \(b\) is in the [[Shell-Thron region]] if the sequence of values

\( \{ b, b^b, b^{b^b}, b^{b^{b^b}}, ... \}\) converge to a finite [[fixed point]].

..

</ref>:

We say that the base \(b\) is in the [[Shell-Thron region]] if the sequence of values

\( \{ b, b^b, b^{b^b}, b^{b^{b^b}}, ... \}\)

converge to a finite [[fixed point]].

==References==

<references/>

[[Category:Exp]]

[[Category:Iterate]]

[[Category:Shell-Thron region]]

[[Category:Tetration]]

[[Category:William Paulsen]]