Volume 3, Issue 6, December 2014, Page: 307-314
Evaluation of Holomorphic Ackermanns
Dmitrii Kouznetsov, Institute for Laser Science, University of Electro-Communications, 1-5-1 Chofugaoka, Chofushi, Tokyo, 182-8585, Japan
Received: Nov. 21, 2014;       Accepted: Dec. 17, 2014;       Published: Dec. 27, 2014
DOI: 10.11648/j.acm.20140306.14      View  2196      Downloads  152
Abstract
Holomorphic extension of the Ackermann function is suggested. Algorithms of evaluation of tetration and pentation are discussed and illustrated with explicit plots and complex maps.
Keywords
Ackermann Function, Superfunction, Tetration, Pentationx
To cite this article
Dmitrii Kouznetsov, Evaluation of Holomorphic Ackermanns, Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314. doi: 10.11648/j.acm.20140306.14
Reference
[1]
http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0001 N.H.Abel. Untersuchung der Functionen zweier unabhängig veränderlicher Gröfsen x und y, wie f(x,y), welche die Eigenschaft haben, dafs f(z,f(x,y)) eine symmetrische Function von z, x und y ist. Journal für die reine und angewandte Mathematik, V.1 (1826) Z.1115.
[2]
http://link.springer.com/article/10.1007%2FBF01459088 Wilhelm Ackermann. Zum Hilbertschen Aufbau der reellen Zahlen. Mathematische Annalen 99, Number 1(1928), Z.118-133.
[3]
http://www.tandfonline.com/doi/abs/10.1080/10652460500422247#.VI7FoBaD5N2 M.H.Hooshmand, August 2006, ”Ultra power and ultra exponential functions”, Integral Transforms and Special Functions, Vol. 17, No. 8, 549-558.
[4]
http://eusebeia.dyndns.org/veblen/veblen On Fast-growing Computable Functions. 07 May 2008.
[5]
http://www.ams.org/journals/mcom/2009-78-267/S0025-5718-09-02188-7/home.html http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation 78 (2009), 1647-1670.
[6]
http://mizugadro.mydns.jp/PAPERS/2009vladie.pdf http://www.ils.uec.ac.jp/~dima/PAPERS/2009vladie.pdf D.Kouznetsov. Tetration as special function. Vladikavkazskii Matematicheskii Zhurnal, 2010, v.12. issue 2, p.31-45.
[7]
https://bbuseruploads.s3.amazonaws.com/bo198214/bunch/downloads/main.pdf Henryk Trappmann, Dmitrii Kouznetsov. 5+ methods for real analytic tetration. June 28, 2010.
[8]
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html https://www.researchgate.net/publication/220576905_Portrait_of_the_four_regular_super-exponentials_to_base_sqrt%282%29 http://eretrandre.org/rb/files/Kouznetsov2009_215.pdf http://www.ils.uec.ac.jp/~dima/PAPERS/2010q2.pdf http://mizugadro.mydns.jp/PAPERS/2010q2.pdf D.Kouznetsov, H.Trappmann. Portrait of the four regular superexponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
[9]
http://link.springer.com/article/10.3103%2FS0027134910010029 https://www.researchgate.net/publication/226675861_Superfunctions_and_sqrt_of_factorial D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12. (Russian version: p.8-14)
[10]
http://link.springer.com/article/10.3103%2FS0027134910020049#page-2 http://www.ils.uec.ac.jp/~dima/PAPERS/2010logistie.pdf http://mizugadro.mydns.jp/PAPERS/2010logistie.pdf D.Kouznetsov. Holomorphic extension of the logistic sequece. Moscow University Physics Bulletin, 2010, series 3, issue 2, page 23-30
[11]
http://link.springer.com/article/10.1007%2Fs00010-010-0021-6 http://www.ils.uec.ac.jp/~dima/PAPERS/2011uniabel.pdf https://www.researchgate.net/publication/226947838_Holomorphic_extension_of_the_logistic_sequence http://mizugadro.mydns.jp/PAPERS/2011uniabel.pdf H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011)
[12]
http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2012-02590-7/S0025-5718-2012-02590-7.pdf https://www.researchgate.net/publication/51892955_Computation_of_the_Two_Regular_Super-Exponentials_to_base_exp%281e%29 http://www.ils.uec.ac.jp/~dima/PAPERS/2012e1eMcom2590.pdf http://mizugadro.mydns.jp/PAPERS/2012e1eMcom2590.pdf H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of Computation, 2012, 81, p.2207-2227.
[13]
http://link.springer.com/article/10.1007/s10043-013-0058-6 http://www.ils.uec.ac.jp/~dima/PAPERS/2012or.pdf http://mizugadro.mydns.jp/PAPERS/2012or.pdf Dmitrii Kouznetsov. Superfunctions for optical amplifiers. Optical Review, July 2013, Volume 20, Issue 4, pp 321-326.
[14]
http://www.m-hikari.com/ams/ams-2013/ams-129-132-2013/kouznetsovAMS129-132-2013.pdf http://www.ils.uec.ac.jp/~dima/PAPERS/2013hikari.pdf http://mizugadro.mydns.jp/PAPERS/2013hikari.pdf D.Kouznetsov. Entire function with logarithmic asymptotic. Applied Mathematical Sciences, 2013, v.7, No.131, p.6527-6541.
[15]
http://mizugadro.mydns.jp/t/index.php?title=Superfunctions_in_Russian http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf http://mizugadro.mydns.jp/BOOK/202.pdf D.Kouznetsov. Superfunctions. LAP, 2014.08.01. ISBN 3659562025, in Russian.
[16]
http://mizugadro.mydns.jp/t/index.php/Abelfunction
[17]
http://mathworld.wolfram.com/AckermannFunction.html
[18]
http://mizugadro.mydns.jp/t/index.php/Superfunction
[19]
http://en.citizendium.org/wiki/Superfunction
[20]
http://mizugadro.mydns.jp/t/index.php/Tetration
[21]
http://en.citizendium.org/wiki/Tetrational
[22]
http://mizugadro.mydns.jp/t/index.php/Transfer_equation
[23]
hrrp://www.ils.uec.ac.jp/~dima/2014ACKER/ Latex version and the C++ generators of figures of this article.
Browse journals by subject