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Solution of $F(z+1)=\exp \big (F(z)\big )$ in complex $z$-plane

Author: Dmitrii Kouznetsov
Journal: Math. Comp. 78 (2009), 1647-1670
MSC (2000): Primary 30A99; Secondary 33F99
Published electronically: January 6, 2009
MathSciNet review: 2501068
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Abstract: Tetration $F$ as the analytic solution of equations $F(z-1)=\ln (F(z))$, $F(0)=1$ is considered. The representation is suggested through the integral equation for values of $F$ at the imaginary axis. Numerical analysis of this equation is described. The straightforward iteration converges within tens of cycles; with double precision arithmetics, the residual of order of 1.e-14 is achieved. The numerical solution for $F$ remains finite at the imaginary axis, approaching fixed points $L$, $L^{*}$ of logarithm ($L=\ln L$). Robustness of the convergence and smallness of the residual indicate the existence of unique tetration $F(z)$, that grows along the real axis and approaches $L$ along the imaginary axis, being analytic in the whole complex $z$-plane except for singularities at integer the $z<-1$ and the cut at $z<-2$. Application of the same method for other cases of the Abel equation is discussed.

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Dmitrii Kouznetsov
Affiliation: Institute for Laser Science, University of Electro-Communications, 1-5-1 Chofugaoka, Chofushi, Tokyo, 182-8585, Japan

Received by editor(s): March 17, 2008
Received by editor(s) in revised form: June 20, 2008
Published electronically: January 6, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.