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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 // Plot of tetrational $f={\rm tet}_b(z)$ 

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]] <!-- [[File:Ausinmap52r5.jpg|260px]] <!--

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]] [[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]]<center>

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

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