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- ....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
- such that, at least for \(b\!>\!1\), is holomorphic at least in \(\{ z \in \mathbb C : \Re(z)\!>\!-2\}\). H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (21 KB (3,175 words) - 23:37, 2 May 2021
- ...Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011)</ref> is holomorphic solution \(F(z)={\rm tet}_b(z)\) of the equations ...map of this function may be interpreted as a hint that the tetrational is holomorphic not only being considered as function of \(z\) at fixed \(b\), but also as14 KB (2,275 words) - 18:25, 30 July 2019
- 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
- [[File:Penplot.jpg|300px|thumb|\(y=\mathrm{pen}(x)\) and related functions.]] ...[hexation]]. The fixed points of pentation are complex, so, for the real–holomorphic superpentation, the method of the [[Cauchi integral]] can be applied, the s7 KB (1,090 words) - 18:49, 30 July 2019
- It is assumed, that \(h\) is [[holomorphic function]] al least in some vicinity of [[halfline]] along the [[real axis] ...y, even the non-integer (and even complex) [[iteration]] of almost every [[holomorphic function]].13 KB (1,766 words) - 18:43, 30 July 2019
- The holomorphic extension of \(h\) is suggested. The holomorphic extension \(F\) of the sequence, generated with such a transfer function, i5 KB (798 words) - 18:25, 30 July 2019
- ...rate''' refer to the [[fractional iterate]] a holomorphic function that is holomorphic in vicinity it its fixed point </ref>. In general, a holomorphic function may have several fixed points, and the fractional iterates, regula20 KB (3,010 words) - 18:11, 11 June 2022
- That publication defined the \(\sqrt{!\,}\) as holomorphic function, ...roduct, the [[SuperFunction]] of Factorial and that for some other special functions are considered there.18 KB (2,278 words) - 00:03, 29 February 2024
- '''Transfer equation''' is relation between some [[holomorphic function]] \(h\), called [[transfer function]] and another function \(F\), ...vial. This determined the interest to the superfunctions from the set of [[holomorphic function]]s of [[complex variable]].3 KB (519 words) - 18:27, 30 July 2019
- [[File:KellerDoyaT.png|300px|thumb|Transfer functions of laser amplifiers with simple kinetics for the short pulses ([[Keller fun H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (11 KB (1,644 words) - 06:33, 20 July 2020
- '''Logarithm''' is [[holomorphic function]], inverse of the [[exponential]]. [[Category:Elementary functions]]4 KB (661 words) - 10:12, 20 July 2020
- H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. [[Aequationes Mathematicae]], v.81, p.65- Abstract — The holomorphic function h is constructed such that h(h(z))=z! ; this function is interpret7 KB (1,091 words) - 23:03, 30 November 2019
- In the simple case, \(f\) is just [[holomorphic function]] of a single variable; then \(L\) is assumed to be a [[complex nu ...] offers the table of evaluations of some fixed points for some elementary functions4 KB (574 words) - 18:26, 30 July 2019
- '''Complex map''' is the graphical representation of a [[holomorphic function]] with the isolines of its real part and those of its imaginary pa For any holomorphic function, in any point, the isolines of the real part are orthogonal to tho2 KB (254 words) - 06:59, 1 December 2018
- [[Category:Holomorphic functions]]5 KB (275 words) - 07:00, 1 December 2018
- '''Holomorphic function''' is concept of the theory of functions of complex variables that refers the the existence of the derivative. Then, function \(f\) is called holomorphic on \(C\).1 KB (151 words) - 21:08, 25 January 2021
- ...proximations for the inverse function, it may have sense to consider it as holomorphic function of the complex argument. [[Category:Mathematical functions]]12 KB (1,754 words) - 18:25, 30 July 2019
- ==Relation to other special functions== According to the Axiom [[TORI]] number 6, the simplest among related functions should be considered as principal, primary. From this axiom, it follows, th27 KB (4,071 words) - 18:29, 16 July 2020
- ==Relation with other functions== '''LambertW''' is [[Holomorphic function]] at the whole complex plane except half-line along the negative p8 KB (1,107 words) - 18:26, 30 July 2019
- ...ng with real numbers. However, the difference become clearly seen is these functions are plotted in the complex plane. ...ermined by the two parameters \(P_{\rm sat}\) and \(t\) through the known functions Doya and Tania. Such a model seems to be applied, in particular, for the [[19 KB (2,778 words) - 10:05, 1 May 2021
- For a given function \(T\), called [[transfer function]], the holomorphic solution \(F\) of [[Transfer equation]] In any pair of holomorphic functions \(F\), \(G\!=\!F^{-1}\),11 KB (1,565 words) - 18:26, 30 July 2019
- Such a \(\varphi\) is assumed to be [[holomorphic function]] for some domain of values of \(z\). </ref> in the middle or 20 century. But the [[real-holomorphic]] solution was not constructed that time.5 KB (750 words) - 18:25, 30 July 2019
- 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
- For example, the existence and uniqueness of the holomorphic [[tetration]] with certain properties is declared as theorem in the first p 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
- Function \(\mathrm {tet}(z)\) is holomorphic in the whole complex plane except the line \(\Re(z)\le -2\). where \(\eta\) is holomorphic periodic function with period unity,14 KB (1,972 words) - 02:22, 27 June 2020
- http://www.springerlink.com/content/u712vtp4122544x4 D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 201 ...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 (6 KB (312 words) - 18:33, 30 July 2019
- ...rithm, the \(J^n f\) may have singularities and cutlines even if \(f\) is holomorphic function. ...ay consider application of the fractional differentiation to some specific functions; for example, the polynomial or the exponential.9 KB (1,321 words) - 18:26, 30 July 2019
- http://www.springerlink.com/content/u712vtp4122544x4 D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 201 ...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 (7 KB (381 words) - 18:38, 30 July 2019
- '''ArcCos''', or '''acos''' is [[holomorphic function]], inverse of [[cos]]. '''ArcCos''', or '''acos''', or '''arccos''' is [[holomorphic function]], inverse of [[cos]].5 KB (754 words) - 18:47, 30 July 2019
- Cih is also real-holomorphic, The notations of this article are not good. Many other similar functions should be described and implemented, anf theu should have similar names. Th8 KB (1,211 words) - 18:25, 30 July 2019
- '''ArcSin''', or '''asign''', or '''arcsin''' is [[holomorphic function]], inverse of [[sin]]; \(f=\arcsin(z)\) is holomorphic solution \(f\) of equation9 KB (982 words) - 18:48, 30 July 2019
- Functions \(~y\!=\!\mathrm{coshc}(x)~\) and \(~y\!=\!\mathrm{coshc}'(x)~\) are shown Coshc is [[holomorphic function]] with the only singularity, namely, pole at zero.4 KB (509 words) - 18:26, 30 July 2019
- Knowledge of these constants simplifies evaluation of functions [[ArcCosc]] and [[ArcCohc]]. ...Cosc]] as it it would be elementary function. It has sense to include such functions as [[Tetration]], [[ArcTetration]], [[SuperFactorial]], [[AbelFactorial]],8 KB (1,137 words) - 18:27, 30 July 2019
- '''ArcCosqq''' is holomorphic function defined with through functions [[acosq]] (or [[ArcCosq]]) expressed with2 KB (216 words) - 18:26, 30 July 2019
- '''Acosc1''' is the holomorphic continuation of function [[ArcCosc]] behind the cut line along the negative Acosc1 is [[real holomorphic function]]; for all the branches numbered in this way, the relation below h6 KB (896 words) - 18:26, 30 July 2019
- The BesselJ0 is real-holomorphic, Along the real axis, BesselJ0 oscillates (like other [[Bessel]] functions). The zeros of are denoted with \(j_{0,n}\); where \(n\) is supposed to be6 KB (913 words) - 18:25, 30 July 2019
- '''BesselK0''' or \(K_0\) is holomorphic function, solution \(f\) of equation3 KB (394 words) - 18:26, 30 July 2019
- The only functions can be differentiated, and the name of this function should be explicitly d The [[Serega function]] is not holomorphic; to, the separate expressions for the real and imaginary parts are used. Th12 KB (1,879 words) - 18:26, 30 July 2019
- // '''serega.cin''' is the numerical [[C++]] implementation of functions // Warining: non-holomorphic functions below!1 KB (265 words) - 15:00, 20 June 2013
- The Serega function is not holomorphic, and, therefore, not differentiable, id est, the derivative depends on the are derivatives of functions \(X\) and \(Y\) with respect to the last argument.5 KB (674 words) - 18:25, 30 July 2019
- ...article [[Logistic sequence]] and links cited there, in particular, the [[Holomorphic extension of the logistic sequence]]. ...quence and its inverse functions can be expressed in terms of [[elementary functions]],7 KB (886 words) - 18:26, 30 July 2019
- [[ArcLogisticSequence]]\(_s\) is holomorphic function, inverse of function [[LogisticSequence]], ...is usually interpreted as operation of multiplication or [[combination of functions]].3 KB (380 words) - 18:25, 30 July 2019
- D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 201 Many superfuncitons for the given transfer function exist; and many Abel functions exist too. That called ArcLogisticSequance seems to be the simplest one. Ac6 KB (817 words) - 19:54, 5 August 2020
- [[WrightOmega]] is holomorphic function, solution \(f\) of equations For this reason, both functions, [[Tania function]] and [[WrightOmega]] are used in [[TORI]].4 KB (610 words) - 10:22, 20 July 2020
- In the strip \(|\Im(z)|<\pi\), functions Keller and Keller\(_0\) are equivalent, \(\mathrm{Keller}(z)=\mathrm{Keller ...sidered also as just a complex variable; so, the Keller is treated as just holomorphic function of complex argument.10 KB (1,479 words) - 05:27, 16 December 2019
- ...ty of the real axis (and, in particular, for real values of the argument), functions [[Shoka function|Shoka]] and [[Shoko function|Shoko]] coincide. The [[Shoko function]] can be expressed through the elementary functions:10 KB (1,507 words) - 18:25, 30 July 2019
- The should be unique, in order to avoid confusion with other functions, while all figures are collected in the same directory for some article or two different functions are used in the same expression.<br>6 KB (899 words) - 18:44, 30 July 2019
- ...in order to distinguish it/him/her from other element of the same set. For functions, the name is especially important because it allows to denote the complicat Usually, the names of functions use the ascii characters, namely, letters and, in exceptional cases, [[cife6 KB (901 words) - 18:27, 30 July 2019
- 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 \(\sin(z)^a~\). <!-- How15 KB (2,495 words) - 18:43, 30 July 2019
