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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https://en.citizendium.org/wiki/index.php?title=Tetration/Bibliography&printable=yes
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.
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