Search results
Create the page "Holomorphic functions" on this wiki! See also the search results found.
- ....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
- ...t it is entire elementary function. The Trappmann function is example of [[holomorphic function]] without [[fixed point]]s, suggested in year 2011 by [[Henryk Tra ...</ref>, it is possible to construct at least one [[superfunction]] for any holomorphic function. Therefore, the consideration of the Trappmann function as [[trans9 KB (1,320 words) - 11:38, 20 July 2020
- In order to follow the descriptions of functions and to reproduce and to modify the figures suggested, the installing of C++ ...enerates the [[contour plot]]s; and in particular, the [[complex map]]s of functions of compex variables4 KB (608 words) - 15:01, 20 June 2013
- Usially, it is assumed that \(s\!=\!T'(0)\), and both, \(T\) and \(g\) are holomorphic at least in some vicinity of zero. </ref>; their functions are pretty different.8 KB (1,239 words) - 11:32, 20 July 2020
- [[Fractional iterate]] is concept used to construct non-integer iterates of functions. For a given function \(~T~\), [[holomorphic function|holomorphic]] in vicinity of its [[fixed point]] \(~L~\), the function2 KB (272 words) - 18:25, 30 July 2019
- ...d that \(K\) is positive real number, id est, \(K>0\), and \(T\) is real–holomorphic, G. Szekeres. Regular iteration of real and complex functions.10 KB (1,627 words) - 18:26, 30 July 2019
- [[AuTra]] is real-holomorphic, \(\mathrm{AuTra}(z^*)=\mathrm{AuTra}(z)^*\) ...uired to plot the camera-ready pictures of the complex maps of the related functions.6 KB (1,009 words) - 18:48, 30 July 2019
- Identifier [[nori]] is used in TORI to denote holomorphic function, that in vicinity of the real axis \(\Re(z\ge 0\)) can be expresse [[nori]] is entire holomorphic function.13 KB (1,759 words) - 18:45, 30 July 2019
- 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
- Let \(f\) be holomorphic function defined at some \(C\in \mathbb C\). Iteration of meromorphic functions.4 KB (630 words) - 18:44, 30 July 2019
- Complex conjugation is not a [[holomorphic function]]. This applies both to functions and to numbers.6 KB (921 words) - 18:46, 30 July 2019
- Walter Bergweiler. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151-188. Both, tet and ate are holomorphic functions; so, the representation above can be used for non-integer \(n\). The expone7 KB (1,161 words) - 18:43, 30 July 2019
- All functions \(P\), \(t^n\) and \(Q\) are defined above, and \(f^n\) is expressed in a w ...nction]] and the [[Abel function]] can be expressed in terms of elementary functions. For many cases, instead of to express the [[superfunction]] through the it13 KB (2,088 words) - 06:43, 20 July 2020
- ...as usually, the additional conditions on the asymptotic behavior of these functions is required in order to make the non-integer iterate unique. [[Holomorphic function]]5 KB (830 words) - 18:44, 30 July 2019
- ...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
- ...e number \(n\) of iteration has no need to be integer. As other holomophic functions, the linear function can be iterated even complex number of times. [[Holomorphic function]],2 KB (234 words) - 18:43, 30 July 2019
- D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. The first ackermann functions have special names:10 KB (1,534 words) - 06:44, 20 July 2020
- For the asymptotic expansions of various functions with the oscillator function, the asymptotic behaviour of as well as the modifications for similar cases, it worth to treat it as holomorphic suction of complex argument \(n\).6 KB (770 words) - 18:44, 30 July 2019
- ...n yen, USA cents and Euro cents, with [[holomorphic function|holomorphic]] functions of time. Various approximations with elementary functions with 3 parameters had been considered. Approximation with ellipses happened18 KB (2,080 words) - 13:48, 1 February 2022
- Aiming the application in the physical science, the real-holomorphic solutions are of special interest. One of them is ArqNem. The branch points for all inverse functions of the [[Nemtsov function]] \(\mathrm {Nem}_q\) are the same.7 KB (1,319 words) - 18:46, 30 July 2019
- [[Base sqrt2]] is article about functions that refer to base \(b=\sqrt{2}\). The logarithm is holomorphic in the complex plane with cut along the negative part of the real axis.3 KB (557 words) - 18:46, 30 July 2019
- ...of function sin are considered. The superfunction SuSin is constructed as holomorphic solution of the transfer equation sin(SuSin(z))=SuSin(z+1). The Abel functi D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2017 KB (1,031 words) - 03:16, 12 May 2021
- ...ion of the first three [[Ackermann function]]s functions and their inverse functions. The first three ackermann functions are3 KB (496 words) - 18:45, 30 July 2019
- ...of regular iteration, while the iterates are regular (id est, holomorphic) functions at least in some vicinity of the fixed point. ...ation''' or '''singular iteration''' could be also used to define the same functions.11 KB (1,715 words) - 18:44, 30 July 2019
- ...\, \exp(-x)\) is integrand, \(f\) is supposed to be smooth (and preferably holomorphic) function, ...efficient in the numerical calculations with approximation of atomic wave functions in spherical coordinates.7 KB (997 words) - 18:44, 30 July 2019
- ==Self-inverse functions== Some functions are their own inverse function.3 KB (444 words) - 18:43, 30 July 2019
- Kori is [[entire function]], it is holomorphic in the whole complex plane. The zero in the denominator in the definition i ...escription and implementation of functions [[kori]] and some other related functions ([[maga]], [[mori]], [[naga]], [[nori]]) appear as [[tool]]s for the descri14 KB (1,943 words) - 18:48, 30 July 2019
- ...function \(~z\! \mapsto\! \ln\!\big(\)[[Factorial]]\((z)\big)~\) , that is holomorphic in the most of the complex \(z\) plane (except \(z\!\le\!-1\)). ...aginary part of the argument, it is, indeed, just combination of these two functions.3 KB (478 words) - 18:43, 30 July 2019
- ...jpg|300px|thumb|\(y\!=\!\mathrm{maga}(x)\) (thick black curve) and related functions]] ...the complex plane. For the last option, the holomorphic properties of the functions involved (and their complex maps) should be analysed.8 KB (1,256 words) - 18:44, 30 July 2019
- ...simple combination of the [[Bessel function]] [[BesselJ0]] with elementary functions: ==Relation with other functions==5 KB (750 words) - 10:00, 20 July 2020
- [[NemBra]] is holomorphic function that expresses values of the [[Nemtsov function]] while two other its inverse functions [[ArcNem]] and [[ArkNem]] gives the unnecessary cut lines of the [[Abel fun4 KB (618 words) - 18:46, 30 July 2019
- ...l \(q\) is presented. For real \(q\) function \(\mathrm{Nem}_q\) is real–holomorphic, id est, \(\mathrm{Nem}_q(z^*)=\mathrm{Nem}_q(z)^*\); this simplifies the c Functions [[SuNem]] and [[AuNem]] are used to express the non-integer iterates of the3 KB (400 words) - 18:48, 30 July 2019
- ...e Nemtsov function are considered, and also some properties of the related functions: ...versions had been assigned (designated) the different names. These inverse functions have different positions of the cuts of the range of holomorphism.14 KB (2,157 words) - 18:44, 30 July 2019
- The additional condition, common for all ackermann functions is assumed, Pentation is holomorphic at least in the part of the complex plane, while the real part of the argum5 KB (803 words) - 18:48, 30 July 2019
- H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 ( ...xponential function so as to produce a real fixed point, allowing a unique holomorphic solution. We then use this solution to find a solution to the unperturbed6 KB (950 words) - 18:48, 30 July 2019
- 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. A4 KB (680 words) - 18:43, 30 July 2019
- ...ay of evaluation of the function. In the most of cases, this refers to a [[holomorphic function]] of a single complex variable. ...lation of the function to other elementary functions or with other special functions, defined earlier; even if the relation is asymptotic or approximate.7 KB (991 words) - 18:48, 30 July 2019
- ...-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
- 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 (1915 KB (2,392 words) - 11:05, 20 July 2020
- H.Trappmann, D.Kouznetsov. Uniqueness of holomorphic Abel functions at a complex fixed point pair Aequationes Mathematicae, v.81, p.65-76 (20117 KB (1,082 words) - 07:03, 13 July 2020
- Assume, some holomorphic function \( f \) is given. Consider holomorphic function \(a \) of two variables such that3 KB (341 words) - 21:45, 21 June 2020
- ...([[Интегральная формула Коши]]) is relation for the holomorphic function that expresses its value at some point through the contour integra For holomorphic function \(F\),2 KB (320 words) - 15:14, 21 July 2020
- In TORI, the [[complex map]]s are used to illustrate [[holomorphic function]]s.2 KB (222 words) - 16:00, 28 May 2021
- ...als happen to be not sufficient for the numerical analysis of mathematical functions. Each of them is real-holomorphic and satisfies equation10 KB (1,491 words) - 18:09, 11 June 2022
- [[Asymmetric bell]] (or «Asymmetric bell function») is kind of functions usable for approximation of shapes of simple peaks, [[Asymmetric bell]] is any 4-parametric real-holomorphic function \(F\) with the following properties:3 KB (396 words) - 11:44, 15 August 2022
