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Tetration
[[Tetration]] (or Tetrational) ${\rm tet}_b$ to base $b \in \mathbb R$, $b\!>1$ ...$b\!>\!1$, is holomorphic at least in $\{ z \in \mathbb C : \Re(z)\!>\!-2\}$.

21 KB (3,175 words) - 23:37, 2 May 2021
Mapping
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html ...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
[[Image:Sqrt(factorial)LOGOintegralLOGO.jpg|200px]] [[File:QexpMapT400.jpg|200px]] <center> $u+\mathrm i v=\sqrt{\exp} (x+\mathrm i y)$</center>

25 KB (3,622 words) - 08:35, 3 May 2021
Место науки в человеческом знании
2. Постепенное изменение генотипа из 48 хр ...ец. Заседание Президиума. 'Наука Урала' No. 2 (830), январь 2003.

111 KB (2,581 words) - 16:54, 17 June 2020
Square root of factorial
...le:SquareRootOfFactorial.png|400px|right|thumb| $y\!=\! x!$ and $y\!=\!\sqrt{!\,}(x)$ verus $x$]] [[Square root of factorial]] (half-iteration of [[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
n/2 , \mathrm{ ~~if~~ } n/2 \in \mathbb N \\ (3n\!+\!1)/2 ~ \mathrm{~~over-vice}

5 KB (798 words) - 18:25, 30 July 2019
Regular Iteration
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html ...H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.

20 KB (3,010 words) - 18:11, 11 June 2022
SuperFactorial
..." is [[superfunction]] of [[factorial]] constructed at its [[fixed point]] 2. The smallest integer larger than 2 (id est 3) is chosen as its value at zero, \(\mathrm{SuperFactorial}(0)=3~\

18 KB (2,278 words) - 00:03, 29 February 2024
Abel function
: $T^2(z)=T(T(z))$ ...H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-175

4 KB (547 words) - 23:16, 24 August 2020
Transfer function
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html ...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
ArcTetration
...verse function of [[tetration]], the ArcTetration $ \mathrm {ate}_b $ to base $ b $ satisfies the relations For base $ b\!=\!\mathrm e \!\approx\! 2.71 $, the natural ArcTetration is presented in figure at right with the [[

7 KB (1,091 words) - 23:03, 30 November 2019
B271az.cc
// showing the [[complex map]] of [[ArcTetration]] to base e. // for(m=-2;m<0;m+=2) {M(-4.6,m-.2) fprintf(o,"(%1d)s\n",m);}

3 KB (529 words) - 14:32, 20 June 2013
Tania function
: $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (2) ~ ~ ~ Tania function has two [[branch point]]s: \(~ -\!2\!\pm\! \mathrm i \pi~$. The position of the [[cut line]]s depends on the r

27 KB (4,071 words) - 18:29, 16 July 2020
Doya function
: $ \displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \mathrm{Doya}(z)= \mathrm{Tania}\!\Big(1+\mathrm{ArcTania}(z)\Big)$ :$t\!=\!2$, id est, $~\mathrm{Doya}^2(x)=\mathrm{Doya}\big(\mathrm{Doya}(x)\big)$

19 KB (2,778 words) - 10:05, 1 May 2021
Table of superfunctions
: $ \!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ G(T(z))=G(z)+1 $ | $\displaystyle \frac{-a^2}{z}$

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 that its Function $\sqrt{\exp}$ should not be confused with

5 KB (750 words) - 18:25, 30 July 2019
Iteration
...T.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]]: $~y\!=\!\mathrm{Factorial~}^{~n}(x)~$ for va

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
Корень из факториала
Символ $\sqrt{\,!\,}$ установлен в качестве эмблемы Физфа

7 KB (381 words) - 18:38, 30 July 2019
Filog.cin
// [[Fixed point]] of [[logarithm]] to base $\exp(z)$ is evaluated with routine complex double Filog(complex double z z_type TaniaS(z_type z){int n; z_type s,t=z+z_type(2.,-M_PI);t*=2./9.; t=I*sqrt(t);

2 KB (258 words) - 10:19, 20 July 2020
Filog
...function]] that expresses the [[fixed point]]s of [[logarithm]] to complex base. for base $b\!=\!\exp(z)$.

4 KB (572 words) - 20:10, 11 August 2020
ArcSinc
: \( \mathrm{Sinc}(z)= 1-\frac{z^2}{6}+\frac{z^4}{120}-\frac{z^6}{5040}+ : \(\!\!\!\!\!\!\!\! \mathrm{ArcSinc}(1\!-\!t)= \sqrt{6 t} \left(

4 KB (563 words) - 18:27, 30 July 2019
Bessel function
$ \!\!\!\!\!\!\!\!\! (1) ~ ~ ~ f''(z)+f'(z)/z+(1-\nu/z^2)f(x) =0$ f(x) \approx x^\nu \left( \frac{2^{-\nu}}{\mathrm{Factorial}(\nu)}+ O(x^2) \right)\)

13 KB (1,592 words) - 18:25, 30 July 2019
F45E.cin
...tion of the [[superfunction]] of the [[exponential]] to base $b\!=\!\sqrt{2}$, constructed at the fixed point $L\!=\!4$. 0.12022125769065893274e-1, 0.45849888965617461424e-2,

1 KB (139 words) - 18:48, 30 July 2019
F45L.cin
...tion of the [[Abel function]] of the [[exponential]] to base $b\!=\!\sqrt{2}$, constructed at the fixed point $L\!=\!4$. -0.587369764200886206e-2, 0.289686728710575713e-2,

2 KB (163 words) - 18:47, 30 July 2019
Schroeder equation
Consider [[logarithm]] to base $~s~$ from both sides of equation (1), assuming that $~s~$ and $~g(z)~ (2) \(~ ~ ~ \log_s\Big(~ g\big( T(z)\big)~\Big) = 1 + \log_s\big(g(z)\big)~$

8 KB (1,239 words) - 11:32, 20 July 2020
Zooming equation
(2) $~ ~ ~ g\big(T(z)\big)= K \, g(z) $ http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html

10 KB (1,627 words) - 18:26, 30 July 2019
File:B271t.png
[[Complex map]] of tetration to base $\mathrm e$, isolines of real and imaginary parts of DB b=sqrt(2);

(1,609 × 1,417 (791 KB)) - 08:30, 1 December 2018
File:E1efig09abc1a150.png
[[Complex map]]s of [[tetration]] $\mathrm{tet}_b$ to base $b\!=\!\sqrt{2}$ , right.

(2,234 × 711 (883 KB)) - 08:34, 1 December 2018
File:Esqrt2iterMapT.png
[[Complex map]] of 1/3 th iteration of the [[exponential]] to [[base sqrt(2)]]. $T(x) = \Big(\sqrt{2}\Big){^z}= \exp_b(z)$

(1,092 × 1,080 (1.36 MB)) - 09:43, 21 June 2013

[[Complex map]] of [[exponential]] to [[base sqrt2]], id est, $b=\sqrt{2}$; $u\!+\!\mathrm i v=\exp_{\sqrt{2}}(x\!+\!\mathrm i y)$

(1,765 × 1,729 (1.15 MB)) - 08:35, 1 December 2018

File:ExpQ2plotT.png

[[Explicit plot]] of [[exponential]] to base $b\!=\!\sqrt{2} \approx 1.414213562373095$ The [[fixed point]]s $L\!=\!2$ and $L\!=\!4$ are solutions of the equation

(2,512 × 1,744 (175 KB)) - 08:35, 1 December 2018

File:Filogbigmap100.png

$\mathrm{Filog}(z)$ expresses the [[fixed point]] of [[logarithm]] to base $b\!=\!\exp(z)$. Another fixed point to the same base can be expressed with

(2,870 × 2,851 (847 KB)) - 08:36, 1 December 2018

File:Filogmap300.png

$\mathrm{Filog}(z)$ expresses the [[fixed point]] of [[logarithm]] to base $b\!=\!\exp(z)$. Another fixed point to the same base can be expressed with

(893 × 897 (292 KB)) - 09:40, 21 June 2013

File:IterEq2plotU.png

[[Explicit plot]] of $c$th [[iteration]] of [[exponential]] to [[base sqrt(2)]] for various values of the number $c$ of iterations. ...tation through the [[superfunction]] $F$ of the exponential to base $\sqrt{2}$, constructed at the fixed point $L\!=\!4$, and the corresponding [[Abel f

(2,944 × 2,944 (986 KB)) - 21:42, 27 September 2013

File:LogQ2mapT2.png

[[Complex map]] of [[logarithm]] to base $b\!=\!\sqrt{2}$; ...range of holomorphism is marked with dashed line. Lines $u\!=\!1$, $u\!=\!2$, $u\!=\!4$, $u\!=\!6$ pass through the integer values at the real axis.

(1,765 × 1,729 (1.43 MB)) - 09:43, 21 June 2013

File:Sqrt2figf45bT.png

...primary expansion of the growing [[superexponential]] to base $b\!=\!\sqrt{2}$ built up at the [[fixed point]] $L\!=\!4$. $T(z)=\exp_{\sqrt{2}}(z)=\Big( \sqrt{2} \Big)^z$

(2,180 × 2,159 (1.01 MB)) - 08:52, 1 December 2018

File:Sqrt2figf45eT.png

...omorphic [[superfunction]] $F$ of the [[exponential]] to base $b\!=\!\sqrt{2}$ built up at its fixed point $L\!=\!4$ with condition $F(0)\!=\!5$. The image is almost Figure 2 (left at the bottom) of publication in [[Mathematics of Computation]]

(2,180 × 2,159 (1.18 MB)) - 08:52, 1 December 2018

File:Sqrt2figL45eT.png

...] of the [[Abel function]] $G$ of the [[exponential]] to base $b\!=\!\sqrt{2}$ constructed at the fixed point $L\!=\!4$ with normalization $G(0)\!=\!5$. ..., H.Trappmnn. Portrait of the four regular super-exponentials to base sqrt(2).

(2,180 × 2,159 (1.07 MB)) - 12:53, 20 July 2020

File:Tet10bxr.jpg
[[Explicit plot]] of [[tetration]] for real values of base $b\!>\!1$. { fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');

(2,491 × 1,952 (236 KB)) - 08:53, 1 December 2018
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
File:QexpMapT400.jpg
[[Complex map]] of function $\sqrt(\exp)= \exp^{1/2}$, [[Halfiteration]] of [[exp]]onential to base $\mathrm e$.

(1,881 × 1,881 (1.83 MB)) - 18:26, 11 July 2013
Superfunctions
https://www.morebooks.de/store/gb/book/superfunctions/isbn/978-620-2-67286-3 [[File:978-620-2-67286-3-full.jpg|440px]]

15 KB (2,166 words) - 20:33, 16 July 2023

File:IterEq2plotT.jpg

[[Explicit plot]] of $n$th [[iteration]] of [[exponential]] to [[base sqrt(2)]] for various values of the number $c$ of iterations. ...tation through the [[superfunction]] $F$ of the exponential to base $\sqrt{2}$, constructed at the fixed point $L\!=\!4$, and the corresponding [[Abel f

(2,922 × 2,922 (1.35 MB)) - 08:38, 1 December 2018

File:Ack3a600.jpg

[[Complex map]] of [[tetration]] to base $b\!=\!\sqrt{2}\!\approx\!1.41$ for(m=-30;m<31;m++){if(m==0){M(m,-10.2)L(m,10.2)} else{M(m,-10)L(m,10)}}

(5,130 × 1,793 (1.09 MB)) - 08:28, 1 December 2018

File:Ack3c600.jpg

[[Complex map]] of [[tetration]] to base $b\!=\!3/2\!=1.5$ for(m=-30;m<31;m++){if(m==0){M(m,-10.2)L(m,10.2)} else{M(m,-10)L(m,10)}}

(5,130 × 1,776 (1.5 MB)) - 08:28, 1 December 2018

File:Ack4c.jpg
[[Complex map]] of [[tetration to Sheldon base]] $b\!=\! cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) );

(5,130 × 1,760 (1.92 MB)) - 08:28, 1 December 2018
File:E1e14z600.jpg
$b=\sqrt{2}$, bottom plot. //b=r=2.71

(3,566 × 6,300 (1.85 MB)) - 08:34, 1 December 2018