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Mathematics of Computation

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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
DOI: https://doi.org/10.1090/S0025-5718-09-02188-7
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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Additional Information

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

DOI: https://doi.org/10.1090/S0025-5718-09-02188-7
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.