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  • Tetration
    [[Tetration]] (or Tetrational) \({\rm tet}_b\) to base \(b \in \mathbb R\), \(b\!>1\)<br> ...\(b\!>\!1\), 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]]<small> [[File:QexpMapT400.jpg|200px]]<br><small><center> \(u+\mathrm i v=\sqrt{\exp} (x+\mathrm i y)\)</center></small>
    25 KB (3,622 words) - 08:35, 3 May 2021
  • Место науки в человеческом знании
    2. <i>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<br> ...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
  • 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
  • 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
  • Base e1e
    ...pg|200px|thumb|Thick green curve: \(y=\eta^x\); thin red curve: \(y=(\sqrt{2})^x\)]] [[Base e1e]] refers to the value of base \(b= \eta =\exp(1/\mathrm e)\approx 1.4446678610\)
    4 KB (559 words) - 17:10, 10 August 2020
  • Base sqrt2
    ...apT.png|300px|thumb|[[Complex map|Map]] of [[exponent]] to base \(b=\sqrt{2}\); lines of constant \(u\) and lines of constant \(v\) show ...00.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
    </ref> and \(~y=\sin^n(\pi/2)-\sin^n(x)~\) with \(~n\!=\!100~\) http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html<br>
    7 KB (1,031 words) - 03:16, 12 May 2021
  • Exotic iteration
    Let \(~ T(z)=z+b z^2 + c z^3+..\) T[z_] = z + b z^2 + c z^3;
    11 KB (1,715 words) - 18:44, 30 July 2019
  • Mathematics of Computation
    http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html<br> ...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
  • SinFT
    ...ansform, refers to the integral transform with kernel \(K(x,y)=\sqrt{\frac{2}{\pi}} \sin(xy)\); \(\displaystyle g(x)=\,\)[[SinFT]]\(f\,(x) \displaystyle =\sqrt{\frac{2}{\pi}} \int_0^\infty \sin(xy) \, f(y) \, \mathrm d y\)
    5 KB (807 words) - 18:44, 30 July 2019
  • Special function
    2. Alternative representations of the function, relation of the function to o And kilogram be 2 lb.
    7 KB (991 words) - 18:48, 30 July 2019
  • Sqrt2f21e.cin
    ...uggests routine F21E for evaluation of [[tetration]] to base \(b\!=\!\sqrt{2}\). //In order to evaluate \(\mathrm{tet}_{\sqrt{2}}(z)\), 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. DB TcL[23]={1., //coeff. of expansion of exp(-q(z+1.2 ...) by powers of (2-F).
    1 KB (145 words) - 18:47, 30 July 2019
  • Sqrt2f23e.cin
    ...r evaluation of real–holomorphic superexponential to base \(b\!=\!\sqrt{2}\). //In order to evaluate \(\mathrm{tet}_{\sqrt{2}}(z)\), the routine should be called as F23E(z)
    2 KB (146 words) - 18:47, 30 July 2019
  • Sqrt2f23l.cin
    ...or evaluation of real–holomorphic abelexponential to base \(b\!=\!\sqrt{2}\). //In order to evaluate \(\mathrm{tet}_{\sqrt{2}}(z)\), 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. 0.12022125769065893274e-1, -0.45849888965617461424e-2,
    1 KB (131 words) - 10:44, 24 June 2020
  • Sqrt2f43l.cin
    ...[Sqrt2f43e.cin]] is routine for evaluarion of abelexponent to base \(\sqrt{2}\) that approach 4 at \(-\infty\) and has value zero at 3. -0.587369764200886206e-2, 0.289686728710575713e-2,
    1 KB (124 words) - 18:46, 30 July 2019
  • Sqrt2f45l.cin
    ...at ecaluates the growing [[Abel function]] of the exponent to base \(\sqrt{2}\) -0.587369764200886206e-2, 0.289686728710575713e-2,
    1 KB (131 words) - 18:47, 30 July 2019
  • Nemtsov function and its iterates
    ...ions. Bulletin (New Series) of the American Mathematical society, v.29, No.2 (1993) p.151-188.</ref><ref name="domsta"> ...l as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
    15 KB (2,392 words) - 11:05, 20 July 2020
  • Hellmuth Kneser
    D.Kouznetsov, H.Trappmann. Superfunctions and sqrt of Factorial. [[Tetration]] to base \( b\) appears as holomorphic solution \( F= \mathrm{tet}_b \)
    12 KB (1,732 words) - 14:01, 12 August 2020
  • Shell-Thron region
    We say that the base \(b\) is in the [[Shell-Thron region]] if the sequence of values We say that the base \(b\) is in the [[Shell-Thron region]] if the sequence of values
    7 KB (1,082 words) - 07:03, 13 July 2020
  • Sqrt2f45e.cin
    // superexponential to base \( \sqrt{2} \) 0.12022125769065893274e-1, 0.45849888965617461424e-2,
    1 KB (112 words) - 13:42, 7 July 2020
  • Precision
    \( N = \log_{10}(100/0.5) = \log_{10}(200) \approx 2.301029995664 \approx 2 \) ===Exercise 2===
    10 KB (1,491 words) - 18:09, 11 June 2022
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