Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Portrait of the four regular super-exponentials to base sqrt(2)HTML articles powered by AMS MathViewer

by Dmitrii Kouznetsov and Henryk Trappmann
Math. Comp. 79 (2010), 1727-1756 Request permission

Abstract:

We introduce the concept of regular super-functions at a fixed point. It is derived from the concept of regular iteration. A super-function F of h is a solution of F(z+1)=h(F(z)). We provide a condition for F being entire, we also give two uniqueness criteria for regular super-functions.

In the particular case h(x)=b$\hat {\phantom {x}}$x we call F super-exponential. h has two real fixed points for b between 1 and e$\hat {\phantom {x}}$(1/e). Exemplary we choose the base b=sqrt(2) and portray the four classes of real regular super-exponentials in the complex plane. There are two at fixed point 2 and two at fixed point 4. Each class is given by the translations along the x-axis of a suitable representative.

Both super-exponentials at fixed point 4—one strictly increasing and one strictly decreasing—are entire. Both super-exponentials at fixed point 2—one strictly increasing and one strictly decreasing—are holomorphic on a right half-plane. All four super-exponentials are periodic along the imaginary axis. Only the strictly increasing super-exponential at 2 can satisfy F(0)=1 and can hence be called tetrational.

We develop numerical algorithms for the precise evaluation of these functions and their inverses in the complex plane. We graph the two corresponding different half-iterates of h(z)=sqrt(2)$\hat {\phantom {x}}$z. An apparent symmetry of the tetrational to base sqrt(2) disproved.

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• Dmitrii Kouznetsov
• Affiliation: Institute for Laser Science, University of Electro-Communications 1-5-1 Chofugaoka, Chofushi, Tokyo, 182-8585, Japan
• Email: dima@ils.uec.ac.jp
• Henryk Trappmann
• Affiliation: Henryk Trappmann, Kameruner Str. 9, 13351 Berlin, Germany
• Email: henryk@pool.math.tu-berlin.de
• Received by editor(s): June 1, 2009
• Received by editor(s) in revised form: August 9, 2009
• Published electronically: February 12, 2010