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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

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