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• Factorial
  ....Stegun: Handbook of Mathematical Functions. 6. Gamma Function and Related Functions (2010) ...r <math>z</math>, the integral can be expressed in terms of the elementary functions.
  27 KB (3,925 words) - 18:26, 30 July 2019
• Tetration
  H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 ( ...e requirements of the [[computational mathematics]] in the rapidly growing functions; if not, the superfunction of tetration, id est, [[pentation]] (\(\mathrm {
  21 KB (3,175 words) - 23:37, 2 May 2021
• Mapping
  H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 ( [[Image:QFactorialQexp.jpg|400px|right|thumb|Complex maps of functions [[square root of factorial]], left, and \(\sqrt{\exp}\), right.]]
  14 KB (2,275 words) - 18:25, 30 July 2019
• Superfunction
  In particular, the [[Ackernann functions]] and [[tetration]] can be interpreted in terms of [[superfunction]]s. ...ns came from the application to the evaluation of fractional iterations of functions.
  25 KB (3,622 words) - 08:35, 3 May 2021
• Square root of factorial
  ...and the [[Abelfactorial]] \(G\), which are the [[superfunction]] and the [[Abel function]] of [[factorial]]: [[Heils Henryk Abel]] and [[Ernst Schröder]] (even in his time, those works were pretty old),
  13 KB (1,766 words) - 18:43, 30 July 2019
• Abel theorem
  In 1824, [[Niels Henrik Abel]] proved the theorem: http://www.cds.caltech.edu/~nair/abel.pdf
  2 KB (320 words) - 18:25, 30 July 2019
• Regular Iteration
  ...he [[Abel function]] is considered as principal, as it allows to deal with functions that have no real fixed points (and, perhaps, no fixed points at all). ...inverting series for $~F~$, or making the asymptotical expansion for the [[Abel equation]], considering $~G(L\!+\!\varepsilon)~$, where $~L~$ is [[fixed po
  20 KB (3,010 words) - 18:11, 11 June 2022
• SuperFactorial
  ...roduct, the [[SuperFunction]] of Factorial and that for some other special functions are considered there. The AbelFactorial satisfies the [[Abel equation]]
  18 KB (2,278 words) - 00:03, 29 February 2024
• Transfer equation
  ==Abel function and Abel equation== .... Within some domain \(D\in \mathbb C\), the Abel function satisfies the [[Abel equation]]
  3 KB (519 words) - 18:27, 30 July 2019
• Abel function
  For some given [[Transfer function]] \(T\), the '''Abel function''' \(G\) is [[inverse function]] of the corresponding [[superfunct The [[Abel equation]] relates the [[Abel function]] \(G\) and the [[transfer function]] \(T\):
  4 KB (547 words) - 23:16, 24 August 2020
• Transfer function
  [[File:KellerDoyaT.png|300px|thumb|Transfer functions of laser amplifiers with simple kinetics for the short pulses ([[Keller fun In [[TORI]], the term usually refers to the [[Transfer equation]], the [[Abel equation]]; the transfer function is assumed to be given function that appe
  11 KB (1,644 words) - 06:33, 20 July 2020
• Abel equation
  ...unction (considered as [[transfer function]]) \(T\) to the corresponding [[Abel function]] \(G\) in the following way: The Abel equation is closely related to the [[transfer equation]] for the [[superfun
  4 KB (598 words) - 18:26, 30 July 2019
• Logarithm
  Logarithm to base \(b\) is [[Abel function]] of operation \(z\mapsto bz\); For such Abel function, the multiplication to constant \(b\) is considered as the [[Trans
  4 KB (661 words) - 10:12, 20 July 2020
• Nest
  ==Iteration of functions== ...ction \(f\) can be expressed through the [[superfunction]] \(F\) and the [[Abel function]] \(G=F^{-1}\):
  3 KB (438 words) - 18:25, 30 July 2019
• ArcTetration
  [[ArcTetration]] is [[Abel function]] of the [[exponential]]. The ArcTetration satisfies the [[Abel equation]]
  7 KB (1,091 words) - 23:03, 30 November 2019
• Mathematica
  simplification of mathematical expressions, solving equations, evaluation of functions, plotting graphics and other things that require some [[mathematics]]. ...implementations for [[tetration]], [[ArcTetration]], [[SuperFactorial]], [[Abel Factorial]], etc.; of order of a dozen coefficients of the asymptotic expan
  12 KB (1,901 words) - 18:43, 30 July 2019
• LambertW function
  ==Relation with other functions== ...n of such names as [[SuZex]] ([[superfunction]] of zex) and [[AbelZex]] ([[Abel function]] of [[zex]]).
  8 KB (1,107 words) - 18:26, 30 July 2019
• Doya function
  ...on]], the [[ArcTania]] function is the [[Abel function]], satisfying the [[Abel equation]] ...ng with real numbers. However, the difference become clearly seen is these functions are plotted in the complex plane.
  19 KB (2,778 words) - 10:05, 1 May 2021
• Table of superfunctions
  ...1}\) is called [[Abel function]] with respect to \(T\); it satisfies the [[Abel equation]] In any pair of holomorphic functions \(F\), \(G\!=\!F^{-1}\),
  11 KB (1,565 words) - 18:26, 30 July 2019
• Square root of exponential
  ...function \(T\) through its [[superfunction]] \(F\) and the corresponding [[Abel function]] \(G=F^{-1}\): is constructed in terms of the [[superfunction]] and the [[Abel function]], and similarly, the [[square root of factorial]]
  5 KB (750 words) - 18:25, 30 July 2019
• Iteration
  Walter Bergweiler. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151-188 For iteration of functions, the same notation is used also by [[Walter Bergweiler]]
  14 KB (2,203 words) - 06:36, 20 July 2020
• Theorem
  H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (
  2 KB (248 words) - 14:33, 20 June 2013
• Natural tetration
  The pair of functions \(\mathrm {tet}\) and \(\mathrm{ate}\) The non–integer iterations of exponential give the class of functions that grow faster than any polynomial but slower than any exponential.
  14 KB (1,972 words) - 02:22, 27 June 2020
• Квадратный корень из факториала
  ...t/u7327836m2850246/ H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 ( [[Category:Mathematical functions]]
  6 KB (312 words) - 18:33, 30 July 2019
• Корень из факториала
  ...t/u7327836m2850246/ H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 ( [[Category:Abel function]]
  7 KB (381 words) - 18:38, 30 July 2019
• ArcLogisticSequence
  ...is usually interpreted as operation of multiplication or [[combination of functions]]. ArcLogisticSequence is [[Abel function]] for the [[Logistic operator]] as the transfer function,
  3 KB (380 words) - 18:25, 30 July 2019
• Logistic operator
  ...[superfunction]]s is called [[Logistic sequence]], and the corresponding [[Abel function]] is called [[ArcLogisticSequence]]. ==Abel function of the Logistic operator==
  6 KB (817 words) - 19:54, 5 August 2020
• ArcShoka
  [[ArcShoka]] function is inverse function of [[Shoka function]] and [[Abel function]] for the [[Keller function]]; ==Various inverse functions==
  3 KB (441 words) - 18:26, 30 July 2019
• ArcKeller
  ...e [[Keller function]]; there exist the explicit representations for these functions through the [[elementary function]]s: ...[[ArcShoka]] and [[Shoka function]]s, as they are [[superfunction]] and [[Abel function]] of the Keller function:
  4 KB (545 words) - 18:26, 30 July 2019
• ArcLambertW
  ...x]]; \(\mathrm{AuZex}=\mathrm{SuZex}^{-1}\). This function satisfies the [[Abel equation]] With [[superfunction]] of [[zex]], called [[SuZex]], and the [[Abel function]], called [[AuZex]], the [[iterations]] of [[zex]] can be expresse
  3 KB (499 words) - 18:25, 30 July 2019
• AuZex
  [[AuZex]] is also [[Abel function]] for the [[transfer function]] \(\mathrm{zex}(z)=z \exp(z)\). Also, [[AuZex]] satisfies the [[Abel equation]]
  6 KB (899 words) - 18:44, 30 July 2019
• Power function
  These functions can be verified with the Mathematica code below: <poem><nomathjax><nowiki> ...ld be applied to the result. Often this appears dealing with trigonometric functions, one writes, for example, \(\sin^a(z)\) instead of \(\sin(z)^a~\). <!-- How
  15 KB (2,495 words) - 18:43, 30 July 2019
• Trappmann function
  ...ered as a serious obstacle at the building-up its [[superfunction]], the [[Abel function]] and the non–integer [[iterate]]s of function Tra. The inverse function, id est, the [[Abel function]] of \(\mathrm{Tra}\), may be called [[Ahe]] or [[ahe]], in a way
  9 KB (1,320 words) - 11:38, 20 July 2020
• Schroeder equation
  ...[[Schroeder function]]s ) are related to the [[superfunction]]s and the [[Abel function]]s. </ref>; their functions are pretty different.
  8 KB (1,239 words) - 11:32, 20 July 2020
• Fractional iterate
  [[Fractional iterate]] is concept used to construct non-integer iterates of functions. ...ior at infinity, or specification of the [[superfunction]] \(F\) and the [[Abel function]] \(G=F^{-1}\), used to construct the fractional iterate \(T^r\) w
  2 KB (272 words) - 18:25, 30 July 2019
• Zooming equation
  G. Szekeres. Regular iteration of real and complex functions. ...various functions can be constructed also with the [[superfunction]] and [[Abel function]], considering the [[transfer equation]] instead of the [[zooming
  10 KB (1,627 words) - 18:26, 30 July 2019
• SuTra
  ...e function of SuTra, id est, [[AuTra]]\(=\)SuTra\(^{-1}\), satisfies the [[Abel equation]] ...n function is important example of function without [[fixed points]]. Such functions were expected to be difficult for construction of the [[fractional iterate]
  9 KB (1,285 words) - 18:25, 30 July 2019
• ArcTra
  ...ing of approximation of solution of algebraic equation with differentiable functions: The agreement functions for the primary approximations app1, app2, app3, app4 described above and f
  10 KB (1,442 words) - 18:47, 30 July 2019
• AuTra
  [[AuTra]] is [[Abel function]] of the [[Trappmann function]], \(\mathrm{tra}(z)=z+\exp(z)\). [[AuTra]] satisfies the Abel equation
  6 KB (1,009 words) - 18:48, 30 July 2019
• Abelpower
  [[Abelpower]] is [[Abel function]] for the [[power function]], id est, transfer function \(T(z)\!=\ ...] function \(F\), the inverse function \(G\!=\!F^{-1}\) is solution of the Abel equation,
  3 KB (470 words) - 18:47, 30 July 2019
• Superlogarithm
  ...ames of routines for elementary functions coincide with the names of these functions. [[Category:Abel function]]
  1 KB (173 words) - 19:31, 30 July 2019
• Superpower
  The two real-holomorphic superpower functions are considered here: For \(a\!=\!2\), the explicit plots of these two functions are shown in figure 1 at right.
  6 KB (903 words) - 18:44, 30 July 2019
• Iterate of exponential
  Walter Bergweiler. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151-188. ...is [[superfunction]] of [[exponent]], and [[Arctetration]] ate, that is [[Abel function]] of exponent, the \(n\)th iteration can be expressed as follows:
  7 KB (1,161 words) - 18:43, 30 July 2019
• Iterate of linear fraction
  All functions \(P\), \(t^n\) and \(Q\) are defined above, and \(f^n\) is expressed in a w and the interse function \(G=F^{-1}\) appears as the [[Abel function]].
  13 KB (2,088 words) - 06:43, 20 July 2020
• Linear fraction
  The corresponding [[Abel function]] \(t=g^{-1}\) can be expressed as follows: With superfunction \(f\) by (16) and [[Abel function]] \(g\) by (17), the iterate of \(t\) can be written in the usual
  5 KB (830 words) - 18:44, 30 July 2019
• Superfunctions
  ...on]]s, [[superfunction]]s, and the non-integer [[iterate]]s of holomorphic functions. Non-integer iterates of holomorphic functions.<br>
  15 KB (2,166 words) - 20:33, 16 July 2023
• Linear fuction
  The corresponding [[Abel function]] can be expressed as follows: the superfunction and the Abel function in the standard way,
  2 KB (234 words) - 18:43, 30 July 2019
• Ackermann function
  The first ackermann functions have special names: ...lysis]] are based on the first 3 [[ackermann functions]] and their inverse functions in various combinations.
  10 KB (1,534 words) - 06:44, 20 July 2020
• AdPow
  [[AdPow]] is specific [[Abelpower]] function, id est, the specific [[Abel function]] for the [[Power function]]. For \(a\!=\!2\), both functions [[AdPow]] and [[AuPow]] are shown in Fig.1
  2 KB (271 words) - 18:43, 30 July 2019
• ArqNem
  The branch points for all inverse functions of the [[Nemtsov function]] \(\mathrm {Nem}_q\) are the same. ...tion of the Nemtsov function happen to be suitable for construction of the Abel function [[AuNem]] of the [[Nemtsov function]] and the corresponding [[iter
  7 KB (1,319 words) - 18:46, 30 July 2019

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