Tetration/Bibliography: Difference between revisions

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imported>Dmitrii Kouznetsov
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Etymology of tetration
Ethimology of tetration
<ref name="good">{{cite journal
<ref name="good">{{cite journal
|author= [[Reuben Louis Goodstein|R.L.Goodstein]]
|author= [[Reuben Louis Goodstein|R.L.Goodstein]]
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Tetration for base <math>b\!=\!\mathrm{e}</math>
Tetration for base <math>b\!=\!\mathrm{e}</math>
<ref name="k">D.Kouznetsov. Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane. [[Mathematics of Computation]], 2008, in press; preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf</ref> and <math>b=2</math>.
<ref name="k">
{{cite journal
|author=D.Kouznetsov.
|title=Solutions of <math>F(z\!+\!1)=\exp(F(z))</math> in the complex <math>z</math>plane.  
|journal=[[Mathematics of Computation]],
|year=2009
|volume=78
|pages=1647-1670
|url= http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
|preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf
|doi=10.1090/S0025-5718-09-02188-7
}}preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf
</ref><ref name="vladie">
{{cite journal
|author=D.Kouznetsov.
|title=Superexponential as special function.
|journal=[[Vladikavkaz Mathematical Journal]]
|year=2009
|url=http://www.ils.uec.ac.jp/~dima/PAPERS/2009vladie.pdf
| volume=12
| issue=2
|pages=31-45
}}
</ref>
 
Uniqueness of the tetration and arctetration at base <math>b\!>\! \exp(1/\mathrm e)</math>
<ref name="uni">
H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. [[Aequationes Mathematicae]], v.81, p.65-76 (2011)
http://www.springerlink.com/content/u7327836m2850246/
http://tori.ils.uec.ac.jp/PAPERS/2011uniabel.pdf
</ref>
 
Superexponentials (and the tetration) to base <math>b\!=\! \exp(1/\mathrm e)</math>
<ref name="e1e">
H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). [[Mathematics of computation]], in preparation, 2011.
http://tori.ils.uec.ac.jp/PAPERS/2011e1e.pdf
</ref>
 
Superexponentials (and the tetration) for the case <math>1\!<\!b\!<\! \exp(1/\mathrm e)</math>, and, in particular, for
<math>b\!=\!\sqrt{2}</math>  
<ref name="q2">
D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). [[Mathematics of Computation]], 2010, v.79, p.1727-1756.
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
http://tori.ils.uec.ac.jp/PAPERS/2010sqrt2.pdf
</ref>


<!--
Linear and piece-vice approximation of tetration.
Linear and piece-vice approximation of tetration.
<ref name="uxp">
<ref name="uxp">
Line 25: Line 69:
Tetration for <math>b\!=\!\mathrm{e}</math>
Tetration for <math>b\!=\!\mathrm{e}</math>
<ref name="k">D.Kouznetsov. Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane. [[Mathematics of Computation]], 2008, in press; preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf</ref>
<ref name="k">D.Kouznetsov. Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane. [[Mathematics of Computation]], 2008, in press; preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf</ref>
!-->


Solutions of equation <math>F(z+1)=\exp(F(z))</math>:
Other solutions of equation <math>F(z+1)=\exp(F(z))</math>:
<ref name="kneser">
<ref name="kneser">
H.Kneser. “Reelle analytische L¨osungen der Gleichung '('(x)) = ex und verwandter Funktionalgleichungen”.
H.Kneser. “Reelle analytische L¨osungen der Gleichung <math> \varphi(\varphi(x)) = e^x</math> und verwandter Funktionalgleichungen”.
Journal f¨ur die reine und angewandte Mathematik, 187 (1950), 56-67.
[[Journal fur die reine und angewandte Mathematik]], 187 (1950), 56-67.
</ref>
</ref>
<ref name="k">D.Kouznetsov. Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane. [[Mathematics of Computation]], 2008, in press; preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf</ref>


Application of tetration <ref>
Application of tetration <ref>
P.Walker. Infinitely differentiable generalized logarithmic and exponential functions. Mathematics
P.Walker. Infinitely differentiable generalized logarithmic and exponential functions. [[Mathematics of computation]], 196 (1991), 723-733.
of computation, 196 (1991), 723-733.
</ref>
</ref>
<ref name="uxp">
<ref name="uxp">
M.H.Hooshmand. ”Ultra power and ultra exponential functions”. Integral Transforms and
M.H.Hooshmand. ”Ultra power and ultra exponential functions”. [[Integral Transforms and Special Functions]] <b>17</b> (8), 549-558 (2006)
Special Functions 17 (8), 549-558 (2006)
</ref>
</ref>
<ref name="a"> W.Ackermann. ”Zum Hilbertschen Aufbau der reellen Zahlen”. Mathematische Annalen
<ref name="a"> W.Ackermann. ”Zum Hilbertschen Aufbau der reellen Zahlen”. [[Mathematische Annalen]]
99(1928), 118-133</ref>
99(1928), 118-133</ref>
<ref name="k2">
<ref name="k2"/>.
D.Kouznetsov. Ackermann functions of complex argument. In preparation;
Preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf
</ref>.


Ackermann Function
Ackermann Function
<ref name="a"> W.Ackermann. ”Zum Hilbertschen Aufbau der reellen Zahlen”. Mathematische Annalen
<ref name="a"> W.Ackermann. ”Zum Hilbertschen Aufbau der reellen Zahlen”. [[Mathematische Annalen]]
99(1928), 118-133</ref>
99(1928), 118-133</ref>
<ref name="k2">
<ref name="k2"/>.
D.Kouznetsov. Ackermann functions of complex argument. In preparation;
Preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf
</ref>.


Additional literature around
About iterations:
<ref>A.Knoebel. ”Exponentials Reiterated.” Amer. Math. Monthly 88 (1981), 235-252.</ref>
<ref>{{cite journal
|author=A.Knoebel
|title=Exponentials Reiterated
|journal=[[Amer. Math. Monthly]]
|volume=88
|year=1981
|pages=235-252
}}</ref>


Wiki-resources related to tetration:<br>
http://www.proofwiki.org/wiki/Definition:Tetration<br>
http://tori.ils.uec.ac.jp/TORI/index.php/Tetration<br>


<references/>
==References==
{{reflist}}

Latest revision as of 10:10, 16 September 2024

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A list of key readings about Tetration.
Please sort and annotate in a user-friendly manner. For formatting, consider using automated reference wikification.

Etymology of tetration [1].

Tetration for base [2].

Tetration for base [3][4]

Uniqueness of the tetration and arctetration at base [5]

Superexponentials (and the tetration) to base [6]

Superexponentials (and the tetration) for the case , and, in particular, for [7]


Other solutions of equation : [8]

Application of tetration [9] [10] [11] [2].

Ackermann Function [11] [2].

About iterations: [12]

Wiki-resources related to tetration:
http://www.proofwiki.org/wiki/Definition:Tetration
http://tori.ils.uec.ac.jp/TORI/index.php/Tetration

References

  1. R.L.Goodstein (1947). "Transfinite ordinals in recursive number theory". Journal of Symbolic Logic 12.
  2. 2.0 2.1 2.2 D.Kouznetsov. Ackermann functions of complex argument. Preprint of the Institute for Laser Science, UEC, 2008. http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf
  3. D.Kouznetsov. (2009). "Solutions of in the complex plane.". Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7. Research Blogging. preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf
  4. D.Kouznetsov. (2009). "Superexponential as special function.". Vladikavkaz Mathematical Journal 12 (2): 31-45.
  5. H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011) http://www.springerlink.com/content/u7327836m2850246/ http://tori.ils.uec.ac.jp/PAPERS/2011uniabel.pdf
  6. H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of computation, in preparation, 2011. http://tori.ils.uec.ac.jp/PAPERS/2011e1e.pdf
  7. D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756. http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html http://tori.ils.uec.ac.jp/PAPERS/2010sqrt2.pdf
  8. H.Kneser. “Reelle analytische L¨osungen der Gleichung und verwandter Funktionalgleichungen”. Journal fur die reine und angewandte Mathematik, 187 (1950), 56-67.
  9. P.Walker. Infinitely differentiable generalized logarithmic and exponential functions. Mathematics of computation, 196 (1991), 723-733.
  10. M.H.Hooshmand. ”Ultra power and ultra exponential functions”. Integral Transforms and Special Functions 17 (8), 549-558 (2006)
  11. 11.0 11.1 W.Ackermann. ”Zum Hilbertschen Aufbau der reellen Zahlen”. Mathematische Annalen 99(1928), 118-133
  12. A.Knoebel (1981). "Exponentials Reiterated". Amer. Math. Monthly 88: 235-252.