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  • Tetration
    Natural [[tetration]] (dashed) and other [[Ackermann function|ackermanns]] [[Tetration]] (or Tetrational) \({\rm tet}_b\) to base \(b \in \mathbb R\), \(b\!>1\)<br>
    21 KB (3,175 words) - 23:37, 2 May 2021
  • Regular Iteration
    '''Regular Iteration''' or '''regular iterate''' refer to the [[fractional iterate]] a holomorphic function that is holomorphic in vi ...d the fractional iterates, regular at different fixed points, have no need to coincide <ref name="sqrt2">
    20 KB (3,010 words) - 18:11, 11 June 2022
  • Transfer function
    ...ansfer equation]], the [[Abel equation]]; the transfer function is assumed to be given function that appear in these equations. The solutions of these eq ...r a given transfer function, the additional restrictions should be applied to make the [[superfunction]] and the [[Abel function]] unique
    11 KB (1,644 words) - 06:33, 20 July 2020
  • B271az.cc
    // showing the [[complex map]] of [[ArcTetration]] to base e. fprintf(o,"46 45 translate\n 10 10 scale\n");
    3 KB (529 words) - 14:32, 20 June 2013
  • Tania function
    ...aniaT.png|350px|right|thumb| Comparison of [[Tania function]] (thin curve) to the [[Shoka function]] (thick curve) for real values of the argument]] ...maginary axis and 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
    ...for comparison of superfunctions \(F\) for various cases, it is convenient to keep relation \(F(0)=1\) (as it is accepted for [[tetration]], which is [[superfunction]] of [[exponential]]); for this reason, the [[T
    19 KB (2,778 words) - 10:05, 1 May 2021
  • Table of superfunctions
    is called [[superfunction]] with respect to \(T\). ...se function, id est, \(G=F^{-1}\) is called [[Abel function]] with respect to \(T\); it satisfies the [[Abel equation]]
    11 KB (1,565 words) - 18:26, 30 July 2019
  • Iteration
    The zeroth iteration of any non-trivial function is supposed to be [[Identity function]]. Similar notations, where the superscript is used to indicate the number of iterate is used also in [[quantum mechanics]] and th
    14 KB (2,203 words) - 06:36, 20 July 2020
  • Natural tetration
    \(\mathrm{tet}(x),\mathrm e^x,10(\mathrm{tet}(x)-(x\!+\!1))\) [[Natural tetration]] is [[tetration]] to base \(\mathrm e=\exp(1)\approx 1.71~\).
    14 KB (1,972 words) - 02:22, 27 June 2020
  • Tetration to Sheldon base
    ...g|500px]]<small> Explicit plot of real and imaginary parts of tetration to base \(b\!=\!s\)</small> [[File:Shelima600.png|500px]]<small>Distribution of tetration to \(b\!=\!s\) along the imaginary axis</small>
    5 KB (707 words) - 21:33, 13 July 2020
  • Iterated Cauchi
    The specific choice of the contour allows to express values f(t) through the left hans side of (9) with the Transfer equ A. integration along the line \(\Re(t)=1\) from \(t = 1−\mathrm i A\) to \(t = 1+\mathrm i A\),<br>
    6 KB (987 words) - 10:20, 20 July 2020
  • SuZex approximation
    ...imations of [[SuZex]] with elementary functions. All the maps are supposed to be displayed in the same scale. Also, it is assumed that the solution \(F=\mathrm{SuZex}\) decays to the stationary point 0 of the transfer function \(T\) by (1) at infinity, e
    14 KB (2,037 words) - 18:25, 30 July 2019
  • F2048ten.inc
    ...[[F2048ten.inc]] defines the approximations of [[tetration]] to base \(b=10\) along the imaginary axis at the interval (-20,20) in 2048 nodes of the qu // perhaps, Aten=20, and NPO =2048; NPO is supposed to be defined in GLxw.
    89 KB (7,127 words) - 18:46, 30 July 2019
  • F4ten.cin
    ...rking directory) and the [[Transfer equation]] for the exponential to base 10 as [[transfer function]]. DB Lten=log(10.);
    2 KB (287 words) - 15:03, 20 June 2013
  • File:B271t.png
    [[Complex map]] of tetration to base $\mathrm e$, isolines of real and imaginary parts of $v\!=\!\Im(f)\!=\!\mathrm {const}$ are plotted; integer values correspond to the thick lines.
    (1,609 × 1,417 (791 KB)) - 08:30, 1 December 2018
  • File:B271t3T.png
    [[Complex map]] of [[tetration]] to base $\mathrm e\approx 2.71$ ...cin]] and [[conto.cin]] should be loaded to the working directory in order to compile the [[C++]] code below
    (1,742 × 1,726 (1,007 KB)) - 08:30, 1 December 2018
  • File:E1efig09abc1a150.png
    [[Complex map]]s of [[tetration]] $\mathrm{tet}_b$ to base<br> This image is close to the figure 9 in the article
    (2,234 × 711 (883 KB)) - 08:34, 1 December 2018
  • File:ExpQ2mapT.png
    [[Complex map]] of [[exponential]] to [[base sqrt2]], id est, $b=\sqrt{2}$; ...the illustration of the application of the method of [[regular iteration]] to construct the [[superfunction]]
    (1,765 × 1,729 (1.15 MB)) - 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:LogQ2mapT2.png
    [[Complex map]] of [[logarithm]] to base $b\!=\!\sqrt{2}$; Line $u\!=\!7$ tries to pass through the points
    (1,765 × 1,729 (1.43 MB)) - 09:43, 21 June 2013
  • File:Tet10bxr.jpg
    [[Explicit plot]] of [[tetration]] for real values of base $b\!>\!1$. should be loaded in order to compile the code below
    (2,491 × 1,952 (236 KB)) - 08:53, 1 December 2018
  • File:TetPlotU.png
    [[Explicit plot]] of [[tetration]] to [[base e]]; $y=\mathrm{tet}(x)$ is shown with thick pink line. ...1$ and $0$, the graphic of tetration $y\!=\!\mathrm{tet}(x)$ looks similar to that of linear function
    (838 × 2,088 (124 KB)) - 08:53, 1 December 2018
  • File:TetSheldonImaT.png
    Few iterations at the [[Iterated Cauchi]] for the [[Tetration to Sheldon base]]. The solution $F$ is supposed to have the specific boundary behavior:
    (4,359 × 980 (598 KB)) - 09:40, 21 June 2013
  • File:Tetsheldonmap03.png
    [[Complex map]] of [[tetration to Sheldon base]] should be loaded in order to compile the code below
    (2,549 × 702 (982 KB)) - 08:53, 1 December 2018
  • File:QexpMapT400.jpg
    [[Halfiteration]] of [[exp]]onential to base $\mathrm e$. ...tory in order to compile the code below. Actually, the last two evaluate [[tetration]] tet and [[arctetration]] ate; routines [[fsexp.cin]] and [[fslog.cin]] ca
    (1,881 × 1,881 (1.83 MB)) - 18:26, 11 July 2013
  • Superfunctions
    In particular, results for [[tetration]], [[arctetration]] and [[iterate]]s of [[exponential]] are presented. ISBN-10: 3659562025
    15 KB (2,166 words) - 20:33, 16 July 2023
  • File:Ack3a600.jpg
    [[Complex map]] of [[tetration]] to base $b\!=\!\sqrt{2}\!\approx\!1.41$ should be loaded to the working directory in order to compile the code below.
    (5,130 × 1,793 (1.09 MB)) - 08:28, 1 December 2018
  • File:Ack3b600.jpg
    [[Complex map]] of [[tetration]] to base $b\!=\!\exp(1/\mathrm e)\!\approx\!1.44$ should be loaded to the working directory in order to compile the code below.
    (5,130 × 1,776 (1 MB)) - 08:28, 1 December 2018
  • File:Ack3c600.jpg
    [[Complex map]] of [[tetration]] to base $b\!=\!3/2\!=1.5$ should be loaded to the working directory in order to compile the code below.
    (5,130 × 1,776 (1.5 MB)) - 08:28, 1 December 2018
  • File:Ack4a600.jpg
    [[Complex map]] of [[tetration]] to base $b\!=\!2$ should be loaded to the working directory in order to compile the code below.
    (5,130 × 1,776 (1.65 MB)) - 11:57, 21 July 2020
  • File:Ack4aFragment.jpg
    [[Complex map]] of [[tetration]] to base $b\!=\!2$ http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf <br>
    (3,457 × 1,776 (1.63 MB)) - 08:28, 1 December 2018
  • File:Ack4b600.jpg
    [[Complex map]] of [[tetration]] to base $b\!=\!2$ should be loaded to the working directory in order to compile the code below.
    (5,130 × 1,760 (1.53 MB)) - 11:59, 21 July 2020
  • File:Ack4bFragment.jpg
    [[Complex map]] of [[tetration]] to base $b\!=\!2$ http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf <br>
    (3,457 × 1,776 (1.62 MB)) - 08:28, 1 December 2018
  • File:Ack4c.jpg
    [[Complex map]] of [[tetration to Sheldon base]] $b\!=\! should be loaded to the working directory in order to compile the code below.
    (5,130 × 1,760 (1.92 MB)) - 08:28, 1 December 2018
  • File:Ack4d.jpg
    [[Complex map]] of [[tetration]] to base 10. $u+\mathrm i v=\mathrm{tet}_{10}(x\!+\!\mathrm i y)$
    (1,282 × 440 (266 KB)) - 08:28, 1 December 2018
  • File:Ack4dFragment.jpg
    [[Complex map]] of [[tetration]] to base 10. $u+\mathrm i v=\mathrm{tet}_{10}(x\!+\!\mathrm i y)$
    (3,457 × 1,776 (1.4 MB)) - 08:28, 1 December 2018
  • File:Analuxp01t400.jpg
    This corresponds to the displacement of the map for unity to the right, along the real axis. In this case, it is easier to guess the asymptotic behaviour of the function (last picture, d) from its p
    (2,083 × 3,011 (1.67 MB)) - 08:29, 1 December 2018
  • File:E1eAuMap600.jpg
    [[Complex map]]s of the [[abel function]] of the [[exponent]] to the [[Henryk base]] fprintf(o,"101 151 translate\n 10 10 scale\n");
    (3,543 × 5,338 (1.57 MB)) - 08:34, 1 December 2018
  • File:E1eghalfm3.jpg
    [[Complex map]] of upper half iterate of exponential to base $\eta=\exp(1/\mathrm e)$ ...ppmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). [[Mathematics of Computation]], v.81 (2012), p. 2207-2227.
    (1,750 × 1,341 (1.24 MB)) - 08:34, 1 December 2018
  • File:E1eSuMap600.jpg
    [[Complex map]]s of two superexponentials to the [[Henryk base]] for [[tetration]] $F\!=\! \mathrm{tet}_\eta$
    (3,377 × 5,055 (1.37 MB)) - 08:34, 1 December 2018
  • File:E1etetma8.jpg
    [[Complex map]] of [[tetration]] to [[base e1e]] [[Tetration]] $F=\mathrm{tet}_{\eta}$ is real-holomorphic solution of equation
    (4,472 × 3,320 (1.69 MB)) - 08:34, 1 December 2018
  • File:E1eti4.jpg
    [[Complex map]] of [[arctetration]] to [[base e1e]], $\eta=\exp(1/\mathrm e)$ fprintf(o,"101 151 translate\n 10 10 scale\n");
    (2,236 × 1,660 (1.02 MB)) - 08:34, 1 December 2018
  • File:Shelima600.png
    Explicit plot of real and imaginary part of [[tetration to Sheldon base]] of imaginary argument. -10.00 2.2202212609 -1.3393771522
    (1,693 × 531 (83 KB)) - 08:51, 1 December 2018
  • File:Shelr80.png
    Explicit plot of [[tetration to Sheldon base]] for real values of the argument. For comparison, the black line shows the graphic for real base $b=1.5$, id est, $y=\mathrm{tet}_{1.5}(x)$
    (1,563 × 1,454 (237 KB)) - 08:51, 1 December 2018
  • File:Shelre60.png
    Explicit plot of [[tetration to Sheldon base]] for real values of the argument. should be loaded in order to compile the code below:
    (1,172 × 1,090 (142 KB)) - 08:51, 1 December 2018
  • File:Sqrt27u.png
    ...k of the approximate symmetry of the [[explicit plot]] of [[tetration]] to base $\sqrt{2}$ in figure http://mizugadro.mydns.jp/t/index.php/File:Sqrt27t.jpg http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br>
    (856 × 507 (50 KB)) - 08:52, 1 December 2018
  • File:Sqrt2atemap.jpg
    [[Complex map]] of [[arctetration]] to base $\sqrt{2}$: http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br>
    (1,758 × 1,741 (723 KB)) - 08:52, 1 December 2018
  • File:Sqrt2diimap80.jpg
    [[Complex map]] of iterate number i of exponent to base $\sqrt{2}$ constructed at its lower ("down") fixed point 2: http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br>
    (2,302 × 2,306 (1.27 MB)) - 08:52, 1 December 2018
  • File:Sqrt2eitet.jpg
    [[iterate]]s of the [[esponent]] to base $\sqrt{2}$, constructed with [[tetration]] and [[arctetration]] to thie base.
    (3,051 × 3,022 (1.36 MB)) - 08:52, 1 December 2018
  • File:Sqrt2q2map600.jpg
    [[Complex map]] of the half iterate of exponent to base $\sqrt{2}$ regular at its lowest fixed point. This function is expressed through [[tetration]] and [[arctetration]] to base $\sqrt{2}$:
    (1,766 × 1,750 (1.43 MB)) - 08:52, 1 December 2018
  • File:Sqrt2srav.png
    Comparison or two half iterate of exponent to base \( \sqrt{2} \) constructed at fixed point 2 and at fixed point 4. scaled with factor \(10^{24} \).
    (2,532 × 1,639 (263 KB)) - 10:53, 24 June 2020
  • File:Sqrt2sufuplot.png
    Four superexponentials to base \(b=\sqrt{2}\) scaled with factor \(10^{24} \) is shown with violet sinusoidal bell.
    (3,520 × 2,507 (408 KB)) - 10:11, 10 June 2022
  • File:Sqrt2tetatemap.jpg
    for the [[tetration]] and [[arctetration]] to base $\sqrt{2}$
    (1,758 × 1,741 (1,008 KB)) - 08:52, 1 December 2018
  • File:Sqrt2tetmap.jpg
    [[Complex map]] of [[tetration]] to base $\sqrt{2}$: http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br>
    (1,758 × 1,741 (656 KB)) - 08:52, 1 December 2018
  • File:Sqrt2uiimap80.jpg
    [[Complex map]] of iterate number i of exponent to base $\sqrt{2}$ constructed at its upper fixed point 4: http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br>
    (2,302 × 2,306 (1.84 MB)) - 08:52, 1 December 2018
  • File:Tet5loplot.jpg
    Graphical search for the real [[fixed point]]s of [[tetration]]: $\mathrm e\!=\!\exp(1)\!\approx\!2.71$ is base of the natural logarithm,
    (1,477 × 1,486 (283 KB)) - 08:53, 1 December 2018
  • File:TetsheldonmapT600.jpg
    [[Complex map]] of [[Tetration]] to the [[Sheldon base]] $b= 1.52598338517 + 0.0178411853321\, \mathrm i$. Constants $L_1$ and $L_2$ are [[fixed point]]s of logarithm to base $b$; they are determined (and evaluated) through function [[filog]]:
    (5,130 × 1,776 (1.02 MB)) - 08:54, 1 December 2018
  • File:Tetsheldonzoo.jpg
    [[Complex map]] of [[tetration to Sheldon base]], zoom-in from the central part of figure should be loaded in order to compile the code below
    (4,337 × 4,317 (1.73 MB)) - 08:54, 1 December 2018
  • Base e1e
    [[Base e1e]] refers to the value of base \(b= \eta =\exp(1/\mathrm e)\approx 1.4446678610\) ...corresponding [[exponential]], [[SuperExponential]] (in particular, the [[tetration]]) and the inverse functions.
    4 KB (559 words) - 17:10, 10 August 2020
  • E1etf.cin
    // [[e1etf.cin]] is routine that evaluates [[tetration to Henryk base]] \(\eta=\exp(1/\mathrm e)\). s[3]= t*(t*(t- 5/ 2.)+ 5/ 2.)- 7/10.;
    2 KB (203 words) - 18:48, 30 July 2019
  • Sqrt2f21e.cin
    // [[Sqrt2f21e.cin]] suggests 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 F
    1 KB (109 words) - 18:48, 30 July 2019
  • Sqrt2f23e.cin
    ...sts routine F21E for 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 F
    2 KB (146 words) - 18:47, 30 July 2019
  • File:Tet2LMap.png
    [[Complex map]] of [[Tetration to base 2]] with "subtracted" singulatiry at \(-2\): ...allows to extent the range of validity of approximation of [[Tetration to base 2]].
    (2,291 × 1,760 (269 KB)) - 19:55, 6 August 2020
  • Tartaria.Math
    ==Example with [[tetration]] to integer base for integer arguments== Is \( Y \) an integer factor of 10?
    3 KB (338 words) - 09:54, 14 January 2020
  • Kuznetsova theorem
    ...tsova theorem]] refers to residual of division of [[tetration]] to integer base by any integer number. Here symbol tet veters to [[tetration]]. The base is indicated as subscript.
    1 KB (150 words) - 20:23, 23 January 2020
  • Place of science
    ...s between science and other kinds of knowledge. This approach is suggested to exclude some concepts from the scientific knowledge by some formal criteria ...research is motivated by huge amount of fake results. Many of them pretend to be scientific.
    101 KB (14,271 words) - 20:58, 25 September 2020
  • File:Tet2uMap.jpg
    approximation of [[tetration to base 2]] for big values of imaginary part of the argument. //Files [[ado.cin]], [[conto.cin]] should be loaded in order to compile the code below.//<pre>
    (1,729 × 1,120 (526 KB)) - 07:21, 24 July 2020
  • File:Tet2LiMap18a.png
    [[Complex map]] of approximation [[Tet2Li]] of [[tetration to base 2]] // c=-log((c-d)/(c+d))/log(10.);
    (1,148 × 1,162 (975 KB)) - 17:42, 28 July 2020
  • File:Tet2LiMap18c.jpg
    [[Complex map]] of approximation [[Tet2Li]] of [[tetration to base 2]] // c=-log((c-d)/(c+d))/log(10.);
    (2,296 × 2,324 (919 KB)) - 17:43, 28 July 2020
  • File:Tet2M2a200.png
    [[Complex map]] of [[Tetration to base 2]]. // c=-log((c-d)/(c+d))/log(10.);
    (2,297 × 1,882 (308 KB)) - 19:59, 6 August 2020
  • File:TetKK200.png
    ...tic behavior of [[Tetration]] to real base \(b\), versus logarithm of this base, \( \beta=\ln(b)\). Values correspond to the upped half of the complex plane; so, \(\Im(L)\ge 0\). Curve for \(L^*\)
    (897 × 1,279 (29 KB)) - 12:45, 12 August 2020
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