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  • Factorial
    :<math> F(z+1)=(z+1) F(z) </math> for all complex <math>z</math> except negative integer values.
    27 KB (3,925 words) - 18:26, 30 July 2019
  • Tetration
    [[File:Ackerplot.jpg|360px]] : \({\rm tet}_b(z\!+\!1) = \exp_b\!\big( {\rm tet}_b(z) \big)\)
    21 KB (3,175 words) - 23:37, 2 May 2021
  • Tetre2215.cc
    // that is convetred to tetreal2215.jpg <br> // Plot of tetrational \(f={\rm tet}_b(z)\)<br>
    6 KB (1,030 words) - 18:48, 30 July 2019
  • Mapping
    [[File:Tetreal2215.jpg|300px|right|thumb|Map of function \(f={\rm tet}_b(x)\) in the \(x,b\) plane ...tsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
    14 KB (2,275 words) - 18:25, 30 July 2019
  • Superfunction
    \(\displaystyle {{F(z)} \atop \,} {= \atop \,} {T^z(t) \atop \,} {= \atop \,}
    25 KB (3,622 words) - 08:35, 3 May 2021
  • Место науки в человеческом знании
    [[File:DimaY2O3ethanol.jpg|260px]] <!--<small> [[File:Ausinmap52r5.jpg|260px]]<small> <!--
    111 KB (2,581 words) - 16:54, 17 June 2020
  • Logiu.cin
    // \(T(z)=4 z (1\!-\!z)\) z_type J(z_type z){ return .5-sqrt(.25-z/Q); }
    3 KB (513 words) - 18:48, 30 July 2019
  • Square root of factorial
    ...le:SquareRootOfFactorial.png|400px|right|thumb| \(y\!=\! x!\) and \(y\!=\!\sqrt{!\,}(x)\) verus \(x\)]] ...[[Factorial]]), or \(\sqrt{!\,}\) is solution \(h\) of equation \(h(h(z))=z!\).
    13 KB (1,766 words) - 18:43, 30 July 2019
  • Holomorphic extension of the Collatz subsequence
    : \(h(F(z))=F(z\!+\!1)\) [[File:File-Sirakuse03a.jpg|right|500px|thumb|Complex map of function \(F\)]]
    5 KB (798 words) - 18:25, 30 July 2019
  • Regular Iteration
    ...tsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756. :$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ F(z\!+\!1)=T(F(z))$
    20 KB (3,010 words) - 18:11, 11 June 2022
  • SuperFactorial
    : \(\mathrm{SuperFactorial}(z)=\mathrm{Factorial}^z(3)\) : \(\mathrm{Factorial}(\mathrm{SuperFactorial}(z))=\mathrm{SuperFactorial}(z\!+\!1)\)
    18 KB (2,278 words) - 00:03, 29 February 2024
  • Transfer function
    D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670. ...tsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
    11 KB (1,644 words) - 06:33, 20 July 2020
  • Nest
    :\(\mathrm{Nest}[f,z,c]\) where \(f\) is name of iterated function, \(z\) is initial value of the argument, and \(c\) is number of iterations.
    3 KB (438 words) - 18:25, 30 July 2019
  • Japan
    [[File:Japan2459014.6416.jpg|320px|right]] [[File:MicroChipAtomicTrap00.jpg|144px]] <small>[[microchip atom trap|Atom trap]] <ref name="nakagawa2006">{
    15 KB (2,106 words) - 13:37, 5 December 2020
  • ArcTetration
    D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation 78 (2009), 1647-1670. : \( \mathrm{ate}_b(\mathrm{tet}_b(z))=z \)
    7 KB (1,091 words) - 23:03, 30 November 2019
  • Tania function
    f'(z)= \frac{f(z)}{1+f(z)} ...nd then along the straight line (parallel to the real axis) to the point \(z\).
    27 KB (4,071 words) - 18:29, 16 July 2020
  • Doya function
    In vicinity of the real axis (While \(|\Im(z)| \!<\! \pi\)), the [[Doya function]] can be expressed through the \mathrm{Doya}(z)=\mathrm{LambertW}\Big( z~ \mathrm{e}^{z+1} \Big)\)
    19 KB (2,778 words) - 10:05, 1 May 2021
  • Doya.cin
    z_type ArcTania(z_type z) {return z + log(z) - 1. ;} z_type ArcTaniap(z_type z) {return 1. + 1./z ;}
    3 KB (480 words) - 14:33, 20 June 2013
  • Table of superfunctions
    : \( \!\!\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ F(z+1)=T(F(z)) \) : \( \!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ G(T(z))=G(z)+1 \)
    11 KB (1,565 words) - 18:26, 30 July 2019
  • Square root of exponential
    [[Square root of exponential]] \(\varphi=\sqrt{\exp}=\exp^{1/2}\) is half-iteration of the [[exponential]], id est, such function tha : \(\!\!\!\!\!\!\!\!\!\!\!\!\!(1) ~ ~ ~ \varphi(\varphi(z))=z\)
    5 KB (750 words) - 18:25, 30 July 2019
  • Iteration
    [[File:PowIteT.jpg|400px|thumb|Fig.1. Iterates of \(T(z)=z^2~\): \(~y\!=\!T^n(x)\!=\!x^{2^n}~\) for various \(n\)]] [[File:FacIteT.jpg|400px|thumb|Fig.2. Iterates of [[Factorial]]: \(~y\!=\!\mathrm{Factorial~}^
    14 KB (2,203 words) - 06:36, 20 July 2020
  • Квадратный корень из факториала
    [[File:SquareRootOfFactorial.png|400px|right|thumb| \(y\!=\! x!\) и \(y\!=\!\sqrt{!\,}(x)\) как функции от \(x\)]] ...из факториала ([[Square root of factorial]]), то есть \(\sqrt{\,!\,}\) - голоморфная функкция \(f\) такая, что
    6 KB (312 words) - 18:33, 30 July 2019
  • Iterated integral
    [[File:Caputo.jpg|200px|right|thumb|M.Caputo]] (id est, \(\sqrt{-\!1}~\) ) in [[Mathematica]] and the [[Identity function]], which is also
    9 KB (1,321 words) - 18:26, 30 July 2019
  • Корень из факториала
    : \(h(h(z))=z\) для некоторого домена значений \(z\).
    7 KB (381 words) - 18:38, 30 July 2019
  • Filog.cin
    ...\(\exp(z)\) is evaluated with routine complex double Filog(complex double z) below z_type ArcTania(z_type z) {return z + log(z) - 1. ;}
    2 KB (258 words) - 10:19, 20 July 2020
  • Filog
    [[File:FilogmT1000a.jpg|300px|right|thumb|\(u\!+\! \mathrm i v=\mathrm{Filog}(x\!+\! \mathrm i y)\) [[File:Figlogzo2t.jpg|300px|right|thumb|\(u\!+\! \mathrm i v=\mathrm{Filog}(x\!+\! \mathrm i y)\)
    4 KB (572 words) - 20:10, 11 August 2020
  • Fac.cin
    z_type fracti(z_type z){ z_type s; int n; DB a[17]= s=a[16]/(z+19./(z+25./(z))); for(n=15;n>=0;n--) s=a[n]/(z+s);
    4 KB (487 words) - 07:00, 1 December 2018
  • ArcSin
    : \(\displaystyle \sin(z) = \frac{\exp(\mathrm i z)- \exp(-\mathrm i z)}{2~ \mathrm i}\) \(f=\arcsin(z)\) is holomorphic solution \(f\) of equation
    9 KB (982 words) - 18:48, 30 July 2019
  • Absorbing Schroedinger
    ...al case, the particle propagates mainly along some coordinate, let it be \(z\), and the potential depends only on the transversal coordinates. ...\!\!\!\!\!\!\!\!\!\! (6) ~ ~ ~ \psi=\mathrm e ^{\mathrm i (c+\mathrm i s) z }\)
    15 KB (2,070 words) - 18:47, 30 July 2019
  • BesselJ0
    J_0''(z)+ J_0'(z)/z+
    6 KB (913 words) - 18:25, 30 July 2019
  • Guiding of waves between absorbing walls
    :\( \displaystyle \!\!\!\!\!\!\!\!\!\! (4) ~ ~ ~ \Psi = \psi ~ \exp(\mathrm i \omega t)\) ...is assumed, and it is supposed that mainly the particle propagates along \(z\) axis, and \(x\) is called the transversal coordinate.!-->
    5 KB (743 words) - 18:47, 30 July 2019
  • BesselK0
    f''(z)+f(z)/z=f(z) K_0(z) = \exp(-z)\sqrt{\frac{\pi}{2z}} ~ \Big( 1+ O(1/z)\Big)
    3 KB (394 words) - 18:26, 30 July 2019
  • Bessel function
    \( \!\!\!\!\!\!\!\!\! (1) ~ ~ ~ f''(z)+f'(z)/z+(1-\nu/z^2)f(x) =0\) Due to singularity of the equation at \(z=0\), the regular solution should have specific behavior. This solution is c
    13 KB (1,592 words) - 18:25, 30 July 2019
  • Efjh.cin
    z_type J(z_type z){ return .5-sqrt(.25-z/Q); } z_type H(z_type z){ return Q*z*(1.-z); }
    3 KB (364 words) - 07:00, 1 December 2018
  • Shoko function
    :\( \!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ \mathrm {Shoko}(z)=\ln\Big(1+\mathrm e^z (\mathrm e -1) \Big)\) ...!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \mathrm {Shoko}(z)= (\mathrm e -1) \mathrm e^z+ O(\mathrm e^{2z})\)
    10 KB (1,507 words) - 18:25, 30 July 2019
  • F45E.cin
    ...valuation of the [[superfunction]] of the [[exponential]] to base \(b\!=\!\sqrt{2}\), constructed at the fixed point \(L\!=\!4\). z_type f45E(z_type z){int n; z_type e,s;
    1 KB (139 words) - 18:48, 30 July 2019
  • Tania.cin
    z_type ArcTania(z_type z) {return z + log(z) - 1. ;} z_type ArcTaniap(z_type z) {return 1. + 1./z ;}
    1 KB (209 words) - 15:01, 20 June 2013
  • Schroeder equation
    [[File:Ernst schroederFragment.jpg|thumb|[[Ernst Schroeder]] (1) \(~ ~ ~ g\Big(T(z)\Big)= s \, g(z)\)
    8 KB (1,239 words) - 11:32, 20 July 2020
  • Zooming equation
    (1) \(~ ~ ~ T\big(f(z)\big)= f( K\, z)~\) (2) \(~ ~ ~ g\big(T(z)\big)= K \, g(z) \)
    10 KB (1,627 words) - 18:26, 30 July 2019
  • ArcTra
    [[File:ArcTraMapT.jpg|200px|thumb| \(u\!+\!\mathrm i v\!=\! \mathrm{ArcTra}(x\!+\!\mathrm i y)~\) (1)\(~ ~ ~ \mathrm{tra}(z)=z+\exp(z)\)
    10 KB (1,442 words) - 18:47, 30 July 2019
  • AuTra.cin
    z_type arctra(z_type z){return z-Tania(z-1.);} z_type tra(z_type z) {return z+exp(z);}
    1 KB (197 words) - 15:03, 20 June 2013
  • Maps of tetration
    [[File:Ack3a600.jpg|400px|thumb|Base \(b=\sqrt{2}\approx 1.41\)]] [[File:Ack3b600.jpg|400px|thumb|Henryk base, \(b=\exp(1/\mathrm e)\approx 1.44\)]]
    5 KB (761 words) - 12:00, 21 July 2020
  • Nori
    [[File:Noriplot300.jpg|400px|thumb|\(y=\mathrm{nori}(x)\) and \(y=100\mathrm{nori}(x)\) ]] [[File:Norifragment.jpg|300px|thumb|\(u\!+\!\mathrm i v=\mathrm{mori}\big(\sqrt{x\!+\!\mathrm i y}\big)^2=\mathrm{nori}(x\!+\!\mathrm i y)\)]]
    13 KB (1,759 words) - 18:45, 30 July 2019
  • Mathematical notations
    ...pplied sequentially, one by one, in raw, for example, <math>A\Big(B\big(C(z)\big)\Big)</math>. :<math> \{ z \in \mathbb{C} : \Im(z)\ge 0 \} </math>.
    5 KB (753 words) - 18:47, 30 July 2019
  • Iterate of linear fraction
    [[File:Fracit05t150.jpg|300px]]\( t(z)=\frac{z}{0.5+z}~\); \(~y=t^n(x)\) versus \(x\) for various \(n\) [[File:Fracit10t150.jpg|300px]]\( t(z)=\frac{z}{1+z} ~\); \(~y=t^n(x)\) versus \(x\) for various \(n\)
    13 KB (2,088 words) - 06:43, 20 July 2020
  • Superfunctions
    [[File:978-620-2-67286-3-full.jpg|440px]] [[File:Ausintay40t50.jpg|240px]]<small><center><p style="line-height:100%">
    15 KB (2,166 words) - 20:33, 16 July 2023
  • Amos
    \(\mathrm{Amos}(z)=\) \(\displaystyle ...ac{1}{2}\mathrm{Lof}(z)-\mathrm{Lof}\Big(\frac{z}{2}\Big)-\frac{\ln(2)}{2} z \right)\)
    6 KB (883 words) - 18:44, 30 July 2019
  • Amplitude of oscillator function
    where \(\psi_n(z)=\frac{1}{\sqrt{N_n}}\)[[HermiteH]]\(_n(z)\, \exp(-z^2/2)\) is [[oscillator function]], [[File:Amoscplot.jpg|360px|right]]
    6 KB (770 words) - 18:44, 30 July 2019
  • Arcsin.cin
    For some reason, the compiler recognises [[exp]], [[log]], [[sin]], [[cos]], of complex argument, but fails with asin and z_type asin0(z_type z){ int m,M; z_type q,s; DB c[41]={
    4 KB (488 words) - 06:58, 1 December 2018
  • Arctran.cin
    z_type tra(z_type z){ return z+exp(z);} z_type arctra1(z_type z){ static DB c[22]={0.,0.5,-0.0625,.005208333333333333, // 0 - 3
    2 KB (354 words) - 06:58, 1 December 2018
  • Base e1e
    ...1emapT1000.jpg|200px|thumb|[[Complex map|Map]] of \(~f\!=\!\eta\!=\!\exp_{\exp(1/\mathrm e)}~\); here \(~u+\mathrm i v=f(x\!+\!\mathrm i y)~\) ]] [[File:Loge1emapT1000.jpg|200px|thumb|[[Complex map|Map]] of \(~f\!=\!\log_{\exp(1/\mathrm e)}~\); here \(~u+\mathrm i v=f(x\!+\!\mathrm i y)~\) ]]
    4 KB (559 words) - 17:10, 10 August 2020
  • Base sqrt2
    ...xpQ2mapT.png|300px|thumb|[[Complex map|Map]] of [[exponent]] to base \(b=\sqrt{2}\); lines of constant \(u\) and lines of constant \(v\) show ...apT1000.jpg|300px|thumb|[[Complex map|Map]] of [[Logarithm]] to base \(b=\sqrt{2}\); lines of constant \(u\) and lines of constant \(v\) show
    3 KB (557 words) - 18:46, 30 July 2019
  • Conjecture on superfunctions
    [[File:Boyt.jpg|360px|thumb|Sledge by http://en.wikipedia.org/wiki/File:Boy_on_snow_sled,_1945.jpg Father of JGKlein. Boy on snow sled, 1945.
    7 KB (1,031 words) - 03:16, 12 May 2021
  • Elementary function
    \(A_{b,n}(z\!+\!1)=A_{b,n-1}\!\big( A_{b,n}(z)\big)\) ..., the real–holomorphism of ackermanns is assumed, \(A_{b,n}(z^*)=A_{b,n}(z)^*\).
    3 KB (496 words) - 18:45, 30 July 2019
  • Exotic iteration
    F(z\!+\!1)=T(F(z)) \( \displaystyle \lim_{z\rightarrow \infty} F(z)=L\)
    11 KB (1,715 words) - 18:44, 30 July 2019
  • Hermite Gauss mode
    [[Hermite Gauss mode]] refers to the specific solution \(F=F(x,z)\) of equation then, assuming some large positive \(M\), expression \(M^2 z/k\) has sense of the coordinate along the propagation of wave, and \(M x/k\
    8 KB (1,216 words) - 18:43, 30 July 2019
  • Hermite polynomial
    [[File:Hermiten.jpg|300px|thumb| Normalised [[Hermite polynomial]]s, \(y=h_n(x)\) for \(n=2,3,4 [[File:Hermiten6map.jpg|312px|thumb| \(u\!+\!\mathrm i v=h_6(x\!+\!\mathrm i y\) ]]
    4 KB (628 words) - 18:47, 30 July 2019
  • Kori
    [[File:Koriplot.jpg|400px|right]] [[File:KorimapFragment.jpg|400px|thumb|\(u\!+\!v=\mathrm{kori}(x\!+\!\mathrm i y)\)]]
    14 KB (1,943 words) - 18:48, 30 July 2019
  • LaguerreL
    TeXForm[TableForm[Table[Table[LaguerreL[n, m, z], {n, 0, 5}], {m, 0, 8}]]] 1 & 1-z & \frac{1}{2} \left(z^2-4 z+2\right) &
    5 KB (759 words) - 18:44, 30 July 2019
  • Maga
    [[File:MagaplotFragment.jpg|300px|thumb|\(y\!=\!\mathrm{maga}(x)\) (thick black curve) and related func \(\mathrm{naga}(x)=\displaystyle 2 \int_0^\infty \mathrm{mori}(p )^2 \exp(\mathrm i p^2 x) \, p \,\mathrm d p\)
    8 KB (1,256 words) - 18:44, 30 July 2019
  • Mathematics of Computation
    D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670. ...tsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
    2 KB (344 words) - 07:02, 1 December 2018
  • Morinaga function
    [[File:MoriplotFragment.jpg|400px|thumb| [[Morinaga function]] and the principal Bessel mode]] [[File:Morimap.jpg|400px|thumb|\(u\!+\!\mathrm i v= \mathrm{mori}(x\!+\!\mathrm i y)\)]]
    15 KB (2,303 words) - 18:47, 30 July 2019
  • Naga
    ...laystyle\mathrm{naga}(p)=2\int_0^\infty \frac{J_0(L_1 x)^2}{(1-x^2)^2} \, \exp(\mathrm i p x^2)\, x\, \mathrm dx\) \(~\displaystyle \mathrm{naga}(p)=2 \int_0^\infty \mathrm{mori}(x)^2 \, \exp(\mathrm i p x^2)\, x\, \mathrm d x\)
    5 KB (750 words) - 10:00, 20 July 2020
  • Oscillator function
    [[File:Hermigaplot.jpg|400px|thumb|\(u\!=\!F_n(x)~\) for \(~n = 0\) .. \(6\)]] [[File:Hermiga6map.jpg|400px|thumb|\(u\!+\!\mathrm i v=F_6(x\!+\!\mathrm i y)\)]]
    6 KB (846 words) - 18:47, 30 July 2019
  • Radial equation for hydrogen atom
    \(z=\sin(\phi)\) \(\Phi(\phi)=\exp(\pm \mathrm i m \phi)\)
    8 KB (1,199 words) - 18:45, 30 July 2019
  • Series
    Series[HankelH1[0, Sqrt[x] ]^2 Pi I Sqrt[x]/2, {x, Infinity, 2}] e^{2 i \sqrt{x}} \left(1-\frac{1}{4} i
    2 KB (325 words) - 18:44, 30 July 2019
  • Sqrt2f21e.cin
    ...n]] suggests routine F21E for evaluation of [[tetration]] to base \(b\!=\!\sqrt{2}\). ...uate \(\mathrm{tet}_{\sqrt{2}}(z)\), the routine should be called as F21E(z)
    1 KB (109 words) - 18:48, 30 July 2019
  • Sqrt2f21l.cin
    // [[Sqrt2f21l.cin]] is code for evaluation of [[arctetration]] to base \(\sqrt{2}\) of complex argument. //z_type tq2L(z_type z){ int n; z_type e,s,k;
    1 KB (145 words) - 18:47, 30 July 2019
  • Sqrt2f23e.cin
    ...1E for evaluation of real–holomorphic superexponential to base \(b\!=\!\sqrt{2}\). ...uate \(\mathrm{tet}_{\sqrt{2}}(z)\), the routine should be called as F23E(z)
    2 KB (146 words) - 18:47, 30 July 2019
  • Sqrt2f23l.cin
    ...23L for evaluation of real–holomorphic abelexponential to base \(b\!=\!\sqrt{2}\). ...uate \(\mathrm{tet}_{\sqrt{2}}(z)\), the routine should be called as F23L(z)
    2 KB (168 words) - 18:47, 30 July 2019
  • Sqrt2f43e.cin
    // [[Sqrt2f43e.cin]] is routine for evaluation of superexponent to base \(\sqrt{2}\) that approach 4 at \(-\infty\) and has value 3 at zero. z_type f43E(z_type z){int n; z_type e,s; DB coeud[23];
    1 KB (131 words) - 10:44, 24 June 2020
  • Sqrt2f45E.cin
    ...te the infinitely growing superfunction of exponential, \(\mathrm{SuExp}_{\sqrt{2},5}\). z_type f45E(z_type z){int n; z_type e,s;
    1 KB (108 words) - 18:47, 30 July 2019
  • SuSin
    ...plot.jpg|300px|thumb| Graphics \(~y\!=\!\mathrm{SuSin}(x)~\) and \(~y\!=\!\sqrt{3/x}~\) ]] [[File:Susinmap.jpg|300px|thumb| \(u+\mathrm i v=\mathrm{SuSin}(x\!+\!\mathrm i y)\)]]
    15 KB (2,314 words) - 18:48, 30 July 2019
  • Суперфункции
    [[File:Covervi.jpg|500px|right|thumb|Обложка (кликается)]] \(T(F(z))=F(z+1)\)
    19 KB (1,132 words) - 20:36, 16 July 2023
  • Nemtsov function and its iterates
    \(~\mathrm{Nem}_q(z)=z+z^3+qz^4 \,\); \(~\) \(\varphi(\varphi(z))\!=\!\exp(z)\). The problem of iteration
    15 KB (2,392 words) - 11:05, 20 July 2020
  • Hellmuth Kneser
    [[File:Hellmuth-Kneser.jpg|200px]]<small> [[File:Qexpmap.jpg|240px]] <small>
    12 KB (1,732 words) - 14:01, 12 August 2020
  • Shell-Thron region
    ...per we will consider the tetration, defined by the equation \( F(z+1)= b^F(z)\) in the complex plane with \( F(0)=1\), for the case where \(b\) is compl ...lso develop iteration methods for numerically approximating the function F(z), both for bases inside and outside the Shell-Thron region. ..
    7 KB (1,082 words) - 07:03, 13 July 2020
  • Sqrt2f45e.cin
    // superexponential to base \( \sqrt{2} \) z_type f45E(z_type z){int n; z_type e,s;
    1 KB (112 words) - 13:42, 7 July 2020
  • Student Distribution
    \frac{\Gamma\left(\frac{\nu+1}{ 2 }\right)}{\sqrt{\pi\ \nu}\ \Gamma\left(\frac{\nu}{2}\right)} \left(1 + \frac{~ t^2}{\nu }\r ...\Gamma\) refers to [[Gamma function]], \(\Gamma(z)\) \(=\) [[Factorial]]\((z\!-\!1)\) .
    12 KB (1,727 words) - 03:21, 11 May 2024
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