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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
• AuNem
  [[AuNem]] is [[Abel function]] of the [[Nemtsov function]], \(\mathrm{Nem}_q(z)=z+z^3+q z^4 ~ ~ \(G=\mathrm{Nem}_q\) is solution of the [[Abel equation]] for the transfer function \(T=\mathrm{Nem}_q\) :
  9 KB (1,441 words) - 18:45, 30 July 2019
• AuPow
  [[AuPow]] 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 (284 words) - 18:43, 30 July 2019
• Base e1e
  ...], [[SuperExponential]] (in particular, the [[tetration]]) and the inverse functions. ...of the [[fixed point]]s and behaviour of the [[superfunction]]s and the [[Abel function]]s are pretty different for the similar values of base \(b\) ment
  4 KB (559 words) - 17:10, 10 August 2020
• Base sqrt2
  [[Base sqrt2]] is article about functions that refer to base \(b=\sqrt{2}\). its [[superfunction]] and the corresponding [[Abel function]].
  3 KB (557 words) - 18:46, 30 July 2019
• Conjecture on superfunctions
  ...(z))=SuSin(z+1). The Abel function AuSin is constructed as solution of the Abel equation AuSin(sin(z))=AuSin(z)+1; in wide range of values z, the relation ...(z))=SuSin(z+1). The Abel function AuSin is constructed as solution of the Abel equation AuSin(sin(z))=AuSin(z)+1; in wide range of values z, the relation
  7 KB (1,031 words) - 03:16, 12 May 2021
• Eigendecomposition
  ...lgorithm for dimensions greater than 4 must either be infinite, or involve functions of greater complexity than elementary arithmetic operations and fractional
  4 KB (518 words) - 18:46, 30 July 2019
• Elementary function
  ...ion of the first three [[Ackermann function]]s functions and their inverse functions. The first three ackermann functions are
  3 KB (496 words) - 18:45, 30 July 2019
• Exotic iteration
  [[Exotic iteration]] refers to the construction of [[superfunction]], the [[Abel function]] and corresponding iterates of the transfer function \(T\) at its ...of regular iteration, while the iterates are regular (id est, holomorphic) functions at least in some vicinity of the fixed point.
  11 KB (1,715 words) - 18:44, 30 July 2019
• NemBra
  ...nctions [[ArcNem]] and [[ArkNem]] gives the unnecessary cut lines of the [[Abel function]] [[Abel function]],
  4 KB (618 words) - 18:46, 30 July 2019
• NemBran
  ...ch, in its turn, necessary for evalution of function [[AuNem]], which is [[Abel function]] for the [[Nemtsov Function]]. Functions [[SuNem]] and [[AuNem]] are used to express the non-integer iterates of the
  3 KB (400 words) - 18:48, 30 July 2019
• Nembran.cin
  ...and [[ArkNem]], but namely [[ArqNem]] is required for evaluation of the [[Abel function]] [[AuNem]] of the [[Nemtsov function]].
  2 KB (281 words) - 07:03, 1 December 2018
• Nemtsov function
  ...e Nemtsov function are considered, and also some properties of the related functions: [[Abel function]], denoted with [[SuNem]],
  14 KB (2,157 words) - 18:44, 30 July 2019
• Pluralism
  H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 ( ...which solution is "best.'' We will approach the problem by first solving Abel's functional equation \(\alpha(e^x) = \alpha(x) + 1\) by perturbing the exp
  6 KB (950 words) - 18:48, 30 July 2019
• SdPow
  ...xpressed as elementary functions. For \(a\!=\!2\), these functions are who functions are shown in Fig.1.
  1 KB (202 words) - 18:48, 30 July 2019
• Sin
  The postulating of some properties of some functions may have sense while no proof is available. Properties of functions [[sin]] and [[cos]], can be deduced from the differential equation above. A
  4 KB (680 words) - 18:43, 30 July 2019
• Special function
  ...lation of the function to other elementary functions or with other special functions, defined earlier; even if the relation is asymptotic or approximate. ...valent of one dollar''' would be suitable in their descriptions as special functions.
  7 KB (991 words) - 18:48, 30 July 2019
• SuNem
  ...mplification of he algorithm, the precision and speed of evaluation of the functions are expected to improve a little bit. Inverse function of [[SuNem]] is function [[AuNem]], that is [[Abel function]] of the [[Nemtsov function]];
  6 KB (967 words) - 18:44, 30 July 2019
• SuPow
  The period \(P=2\pi\mathrm i / \ln(a)\) is the same for both these functions. Functions [[SuPow]] and [[SdPow]] are related with expression
  3 KB (405 words) - 18:43, 30 July 2019
• SuSin
  ...-4/S0273-0979-1993-00432-4.pdf Walter Bergweiler. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151-188</ref>. SuSin can be aproximated with elementary functions.
  15 KB (2,314 words) - 18:48, 30 July 2019
• Nemtsov function and its iterates
  G. Szekeres. Regular iteration of real and complex functions. Acta Mathematica, September 1958, Volume 100, Issue 3, pp 203-258. W.Bergweiler. Iteration of meromorphic functions. Bulletin (New Series) of the American Mathematical society, v.29, No.2 (19
  15 KB (2,392 words) - 11:05, 20 July 2020
• Shell-Thron region
  H.Trappmann, D.Kouznetsov. Uniqueness of holomorphic Abel functions at a complex fixed point pair Aequationes Mathematicae, v.81, p.65-76 (2011
  7 KB (1,082 words) - 07:03, 13 July 2020