Tetration/Bibliography: Difference between revisions

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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 $b\!=\!2$ ^[2].

Tetration for base $b\!=\!\mathrm {e}$ ^[3]^[4]

Uniqueness of the tetration and arctetration at base $b\!>\!\exp(1/\mathrm {e} )$ ^[5]

Superexponentials (and the tetration) to base $b\!=\!\exp(1/\mathrm {e} )$ ^[6]

Superexponentials (and the tetration) for the case $1\!<\!b\!<\!\exp(1/\mathrm {e} )$ , and, in particular, for $b\!=\!{\sqrt {2}}$ ^[7]

Other solutions of equation $F(z+1)=\exp(F(z))$ : ^[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

↑ R.L.Goodstein (1947). "Transfinite ordinals in recursive number theory". Journal of Symbolic Logic 12.
↑ ^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 Cite error: Invalid <ref> tag; name "k2" defined multiple times with different content Cite error: Invalid <ref> tag; name "k2" defined multiple times with different content
↑ D.Kouznetsov. (2009). "Solutions of $F(z\!+\!1)=\exp(F(z))$ in the complex $z$ 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
↑ D.Kouznetsov. (2009). "Superexponential as special function.". Vladikavkaz Mathematical Journal 12 (2): 31-45.
↑ 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
↑ 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
↑ 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
↑ H.Kneser. “Reelle analytische L¨osungen der Gleichung $\varphi (\varphi (x))=e^{x}$ und verwandter Funktionalgleichungen”. Journal fur die reine und angewandte Mathematik, 187 (1950), 56-67.
↑ P.Walker. Infinitely differentiable generalized logarithmic and exponential functions. Mathematics of computation, 196 (1991), 723-733.
↑ M.H.Hooshmand. ”Ultra power and ultra exponential functions”. Integral Transforms and Special Functions 17 (8), 549-558 (2006)
↑ ^11.0 ^11.1 W.Ackermann. ”Zum Hilbertschen Aufbau der reellen Zahlen”. Mathematische Annalen 99(1928), 118-133
↑ A.Knoebel (1981). "Exponentials Reiterated". Amer. Math. Monthly 88: 235-252.

[good-1] R.L.Goodstein (1947). "Transfinite ordinals in recursive number theory". Journal of Symbolic Logic 12.

[k2-2] 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 Cite error: Invalid <ref> tag; name "k2" defined multiple times with different content Cite error: Invalid <ref> tag; name "k2" defined multiple times with different content

[k-3] D.Kouznetsov. (2009). "Solutions of $F(z\!+\!1)=\exp(F(z))$ in the complex $z$ 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

[vladie-4] D.Kouznetsov. (2009). "Superexponential as special function.". Vladikavkaz Mathematical Journal 12 (2): 31-45.

[uni-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

[e1e-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

[q2-7] 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

[kneser-8] H.Kneser. “Reelle analytische L¨osungen der Gleichung $\varphi (\varphi (x))=e^{x}$ 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.

[uxp-10] M.H.Hooshmand. ”Ultra power and ultra exponential functions”. Integral Transforms and Special Functions 17 (8), 549-558 (2006)

[a-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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@@ Line 1: / Line 1: @@
 {{subpages}}
+Etymology of tetration
-Ethimology of tetration
 <ref name="good">{{cite journal
 |author= [[Reuben Louis Goodstein|R.L.Goodstein]]
@@ Line 15: / Line 14: @@
 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>
+<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.
 <ref name="uxp">
@@ Line 25: / Line 69: @@
 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>
+!-->
-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">
-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 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>
-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 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 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]]
 (1928), 118-133</ref>
 <ref name="k2">
-D.Kouznetsov. Ackermann functions of complex argument. In preparation;
+D.Kouznetsov. Ackermann functions of complex argument. Preprint ILS, 2008.  http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf
-Preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf
 </ref>.
 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]]
 (1928), 118-133</ref>
 <ref name="k2">
-D.Kouznetsov. Ackermann functions of complex argument. In preparation;
+D.Kouznetsov. Ackermann functions of complex argument.
-Preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf
+Preprint ILS, 2008. 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}}

Tetration/Bibliography: Difference between revisions

Latest revision as of 21:17, 2 November 2013

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