Difference between revisions of "File:Sqrt2srav.png"

Fig.16.8 from page 230 of book «Superfunctions»^[1], 2020.

It appears also as Рис.16.9 at page 234 of the Russian version «Суперфункции»^[2], 2015

This image is used also in article ^[3] at Mathematics of Computation, 2010.

The figure shows the comparison or two half iterate of exponential to base \( \sqrt{2} \) constructed at fixed point 2 and at fixed point 4.

\( y=\exp_{\sqrt{2},4}^{~1/2}(x) \) , solid line

\( y=\exp_{\sqrt{2},2}^{~1/2}(x) \) , dashed line

The second number in the subscript indicates the fixed point used to perform the regular iterate for the evaluation.

The thin curve represents the difference, \(y=D(x)= \exp_{\sqrt{2},\mathrm 4}^{~1/2}(x) -\exp_{\sqrt{2},\mathrm 2}^{~1/2}(x)\) , scaled with factor \(10^{24} \).

Description

The half iterate of a transfer function \(T\) is such function \(\varphi\) that \[ \varphi(\varphi(z))=T(z) \] It is special case of the \(n\)th iterate of the transfer function with \(n=1/2\).

For \(T=\exp\), such an function had been announced in 1950 by Hellmuth Kneser ^[4] in 1950 and implemented ^[5][6] in 2009, 2010; it satisfies equation \(\varphi(\varphi(z))=\mathrm e^z\).

For general case, the \(n\) iterate of a transfer function \(T\) is expressed through its superfunction \(F\) and its abelfunction \(G=F^{-1}\) as follows: \[ T^n(z)=F(n+G(z)) \] Here, number \(n\) of the iterate has no need to be integer.

The picture corresponds to \(T(z)=\exp_{\sqrt{2}}(z)=\exp(z/2) \).

The superfunction is not unique. The two growing real-holomorphic superexponentials are used:

\(F=\mathrm{SuExp}_{\sqrt{2},1}=\mathrm{tet}_{\sqrt{2}}\) and

\(F=\mathrm{SuExp}_{\sqrt{2},3}\)

The second number in the subscript indicates the value of the superfunction at zero.

For real values of the argument, the first superexponential (tetration base \(\sqrt{2}\)) gives the halfiterate \(T^{1/2}(x)\) that is valid for \(x<4\). The second superexponential gives the halfiterate \(T^{1/2}(x)\) that is valid for \(x>2\).

In the range \(2<x<4\), these two half iterates look very similar; the deviation is smaller than \(10^{-24}\).
In order to show this deviation in the picture, it is scaled with factor \(10^{24}\).

Feel free to look at the book cited for the more detailed description and/or download the code below to reproduce and/or to modify the picture.
At the reuse, attribute the source and indicate the modification(s) in the code, if any.

Please communicate Editor if you see any misprint, mistake, error in the Book, in the picture and/or in its description.

C++ generator of the curves

/* Files ado.cin, sqrt2f23e.cin, sqrt2f23l.cin, sqrt2f43e.cin, sqrt2f43l.cin should be loaded in order to compile the code below.*/

 #include <math.h>
 #include <stdio.h>
 #include <stdlib.h>
 #define DB double
 #define DO(x,y) for(x=0;x<y;x++)
 #include <complex>
 typedef std::complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
// #include "conto.cin"
#include "ado.cin"
 #include "sqrt2f23E.cin"
 #include "sqrt2f23L.cin"
 #include "sqrt2f43E.cin"
 #include "sqrt2f43L.cin"
// #include "superex.cin"
// #include "slog14128.cin"

// #include "exq.cin"
z_type exq(z_type z){   DB T4= 19.236149042042854712;
                        DB T2=-17.143148179354847104;   z_type z3=z-3.;
//      return 2.47e-25*(z-2.)*(4.-z)*(1.+z3*(.122+z3*.009))*sin(0.75+z3*(-.0767+z3*(+.01) )+ T4*log(4.-z) + T2*log(z-2.) );
//      return 2.47e-25*(z-2.)*(4.-z)*(1.+z3*(.122+z3*.009))*sin(0.75+z3*(-.075+z3*(+.01) )+ T4*log(4.-z) + T2*log(z-2.) );
//      return 2.47e-25*(z-2.)*(4.-z)*(1.+z3*(.122+z3*.006))*sin(0.74+z3*(-.07+z3*(+.01) )+ T4*log(4.-z) + T2*log(z-2.) );
//      return 2.48e-25*(z-2.)*(4.-z)*(1.+z3*(.120+z3*.006))*sin(0.747+z3*(-.069+z3*(+.007) )+ T4*log(4.-z) + T2*log(z-2.) );

//      return 2.48e-25*(z-2.)*(4.-z)*(1.+z3*(.120+z3*.006))*sin(0.747+z3*(-.068+z3*(+.007) )+ T4*log(4.-z) + T2*log(z-2.) );
//      return 2.48e-25*(z-2.)*(4.-z)*(1.+z3*(.120+z3*.006))*sin(0.4+z3*(-.067+z3*(-.8) )+ T4*log(4.-z) + T2*log(z-2.) );
//      return 2.48e-25*(z-2.)*(4.-z)*(1.+z3*(.120+z3*.006))*sin(0.4+z3*(-.067+z3*(-.8) )+ T4*log(4.-z) + T2*log(z-2.) );
//      return 2.48e-25*(z-2.)*(4.-z)*(1.+z3*(.120+z3*.006))*sin(0.4+z3*(-.07+z3*(-.8) )+ T4*log(4.-z) + T2*log(z-2.) );
        return 2.48e-25*(z-2.)*(4.-z)*(1.+z3*(.120+z3*.006))*sin(0.4+z3*(-.08+z3*(-.8) )+ T4*log(4.-z) + T2*log(z-2.) );
                }


DB XI[200]={
2.00, 2.01, 2.02, 2.03, 2.04, 2.05, 2.06, 2.07, 2.08, 2.09, 2.10, 2.11, 2.12, 2.13, 2.14, 2.15, 2.16, 2.17, 2.18, 2.19,
2.20, 2.21, 2.22, 2.23, 2.24, 2.25, 2.26, 2.27, 2.28, 2.29, 2.30, 2.31, 2.32, 2.33, 2.34, 2.35, 2.36, 2.37, 2.38, 2.39,
2.40, 2.41, 2.42, 2.43, 2.44, 2.45, 2.46, 2.47, 2.48, 2.49, 2.50, 2.51, 2.52, 2.53, 2.54, 2.55, 2.56, 2.57, 2.58, 2.59,
2.60, 2.61, 2.62, 2.63, 2.64, 2.65, 2.66, 2.67, 2.68, 2.69, 2.70, 2.71, 2.72, 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79,
2.80, 2.81, 2.82, 2.83, 2.84, 2.85, 2.86, 2.87, 2.88, 2.89, 2.90, 2.91, 2.92, 2.93, 2.94, 2.95, 2.96, 2.97, 2.98, 2.99,
3.00, 3.01, 3.02, 3.03, 3.04, 3.05, 3.06, 3.07, 3.08, 3.09, 3.10, 3.11, 3.12, 3.13, 3.14, 3.15, 3.16, 3.17, 3.18, 3.19,
3.20, 3.21, 3.22, 3.23, 3.24, 3.25, 3.26, 3.27, 3.28, 3.29, 3.30, 3.31, 3.32, 3.33, 3.34, 3.35, 3.36, 3.37, 3.38, 3.39,
3.40, 3.41, 3.42, 3.43, 3.44, 3.45, 3.46, 3.47, 3.48, 3.49, 3.50, 3.51, 3.52, 3.53, 3.54, 3.55, 3.56, 3.57, 3.58, 3.59,
3.60, 3.61, 3.62, 3.63, 3.64, 3.65, 3.66, 3.67, 3.68, 3.69, 3.70, 3.71, 3.72, 3.73, 3.74, 3.75, 3.76, 3.77, 3.78, 3.79,
3.80, 3.81, 3.82, 3.83, 3.84, 3.85, 3.86, 3.87, 3.88, 3.89, 3.90, 3.91, 3.92, 3.93, 3.94, 3.95, 3.96, 3.97, 3.98, 3.99};

//seq(10000000000000000000000000 eq42i[i], i = 1 .. 199) // multiply the table below with 10^25
DB DI[200]={0.,
 0.4784698105   -0.05393190341, -0.1300677822,  -0.03171912709, 0.1785649091, 
-0.2250109272,  0.1825902914,   0.03480286362,  -0.3467282069,  0.3008735706, 
 0.2712188539,  -0.4190409114,  -0.3490350151,  0.3694646591,   0.5766896417, 
-0.02558065033, -0.6509275876,  -0.5677290067,  0.1249427107,   0.7445345668, 
 0.7404203400,  0.1276956912,   -0.6160794452,  -0.9628063538,  -0.6916022303, 
 0.01864447876, 0.7488833086,   1.101226313,    0.9014321358,   0.2561118946, 
-0.5300917829,  -1.110965026,   -1.247875232,   -0.8932436694,  -0.1889188837, 
 0.6042806729,  1.208925529,    1.425359378,    1.188972596,    0.5785454907, 
-0.2192350209,  -0.9722698076,  -1.471223669,   -1.584225579,   -1.285665934, 
-0.6552228513,  0.1488558485,   0.9326874399,   1.514042501,    1.762841619, 
 1.626741407,   1.137951659,    0.4018211902,   -0.4286446672,  -1.185893868, 
-1.721405217,   -1.933299208,   -1.783360064,   -1.301259878,   -0.5762799943, 
 0.2610215297,  1.063258680,    1.692178857,    2.041693575,    2.054394851, 
 1.729315148,   1.120250436,    0.3254866584,   -0.5289731871,  -1.309749251,
-1.896727084,   -2.200905751,   -2.176957263,   -1.828915204,   -1.208481545, 
-0.4065111531,  0.4608414413,   1.269162309,    1.903518088,    2.274423794, 
 2.329910221,   2.062170481,    1.508044374,    0.7434411090,   -0.1274043434, 
-0.9862966412,  -1.717093805,   -2.221232577,   -2.430741451,   -2.317083136, 
-1.894734068,   -1.219099562,   -0.3790914522,  0.5146293821,   1.344202005, 
 2.000019677,   2.395098433,    2.476513136,    2.232427985,    1.693805806, 
 0.9305534145,  0.04256669434,  -0.8532073059,  -1.638054138,   -2.207164096,
-2.483657197,   -2.429103975,   -2.049103493,   -1.393092023,   -0.5482898657, 
 0.3715392444,  1.241138307,    1.940691601,    2.372404168,    2.474456697, 
 2.230337845,   1.672108541,    0.8769280448,   -0.04292211556, -0.9553576726, 
-1.727182735,   -2.243596738,   -2.425705096,   -2.243353388,   -1.721136622, 
-0.9363222889,  -.008568273516, 0.9174331778,   1.694075610,    2.194493483, 
 2.334032812,   2.085879418,    1.487850807,    0.6385838664,   -0.3169989468, 
-1.211215472,   -1.882663464,   -2.205965842,   -2.116798723,   -1.627093624, 
-0.8266252487,  0.1305785338,   1.053731641,    1.752439170,    2.076719446, 
 1.951641871,   1.398412552,    0.5356620174,   -0.4416412943,  -1.303325066, 
-1.837659684,   -1.905696400,   -1.482410042,   -0.6719601817,  0.3107105811, 
 1.191600113,   1.712573743,    1.710469692,    1.176345414,    0.2724209897, 
-0.7061642661,  -1.421131980,   -1.609137760,   -1.188763710,   -0.3119998189, 
 0.6722631328,  1.345399787,    1.398988810,    0.7938478007,   -0.1863084618, 
-1.040543314,   -1.297193305,   -0.7929908943,  0.1807419873,   1.000006240, 
 1.091699572,   0.3698400463,   -0.6124347228,  -1.024247436,   -0.4742694444, 
 0.5137381025,  0.8869221813,   0.1975127230,   -0.6902438791,  -0.5357760631, 
 0.4270413342,  0.5566459653,   -0.3748406966,  -0.3814257507,  0.4878409640,
-0.08114350401, -0.2447048034,  0.3328830351,   -0.2984357129,  0.1840035673, 
 0.1664529435,  2.207054268,    341.6955295,    1.549459936e6 };
// I think, data are from Maple.
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
//FILE *o;o=fopen("fig09c.eps","w");  ado(o,124,82);
//  FILE *o;o=fopen("figexcr.eps","w");  ado(o,0,0,126,110);
//  FILE *o;o=fopen("sqrt26pl.eps","w");  ado(o,1260,810);
//  FILE *o;o=fopen("sqrt26sra.eps","w");  ado(o,660,210);
  FILE *o;o=fopen("sqrt2sra.eps","w");  ado(o,1240,810);
// fprintf(o,"62 12 translate\n 10 10 scale\n");
// fprintf(o,"64 30 translate\n 10 10 scale\n");
// fprintf(o,"620 120 translate\n 100 100 scale\n");
 fprintf(o,"620 110 translate\n 100 100 scale\n");
#define M(x,y) fprintf(o,"%7.4f %7.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%7.4f %7.4f L\n",0.+x,0.+y);
//#define o(x,y) fprintf(o,"%7.4f %7.4f o\n",0.+x,0.+y);

M(0,-1.1)L(0,6.6)
M(-6.1,0)L(6.1,0)
fprintf(o,".03 W S\n");
for(m=-6;m<7;m++){if(m!=0){M(m,-1)L(m,6)}}
for(n=-1;n<7;n++){if(n!=0){M(-6,n)L(6,n)}}
fprintf(o,".01 W S\n");

fprintf(o,"1 setlinejoin 0 setlinecap\n");

for(m=2;m<196;m++){ if(m==2) M(XI[m],.1*DI[m]) else L(XI[m],.1*DI[m]) } 
fprintf(o,"1 0 1 RGB S\n");
// No way to see any deviation at this resolution


for(m=0;m<42;m+=1){ x=2.001+.1*m; z=z_type(x,0.); 
//                      d=UQ2L(z); t=Re(d);  y=Re(UQ2E(.5+d));
                        d=F43L(z); t=Re(d);  y=Re(F43E(.5+d));
                //      printf("%6.3f %6.3f %6.3f\n",x,t,y); 
                        if(m==0) M(x,y) else L(x,y)}
fprintf(o,".03 W 0 0 1 RGB S\n");

for(m=0;m<104;m+=1){x=3.99-.1*m; z=x;
        //      z=TQ2L(z);y=Re(TQ2E(.5+z));
                z=F23L(z);y=Re(F23E(.5+z));
                 if(m/2*2==m) M(x,y) else L(x,y)}
fprintf(o,".06 W 1 0 0 RGB S\n");

DB T4= 19.236149042042854712;
DB T2=-17.143148179354847104;
for(m=0;m<500;m+=1){ x=2.002+.004*m; z=z_type(x,0.);
//                       c=UQ2L(z); p=Re(UQ2E(.5+c));
//                       d=TQ2L(z); q=Re(TQ2E(.5+d));
                        c=exq(z);  y=1.e24*Re(c);if(m==0) M(x,y) else L(x,y)}
fprintf(o,".005 W 0 .1 0 RGB S\n");

/* NATURAL SEXP
  // solid thin
DO(m,98){x=-6.2+.1*m; z=z_type(x,0.); d=FSLOG(z);y=Re(FSEXP(.5+d));  if(m==0) M(x,y) else L(x,y)}
fprintf(o,".01 W 0 0 0 RGB S\n");
*/

 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
//      system(      "gv figexcr.eps &"); //for UNIX
//      system(    "open figexcr.eps"); //for macintosh
        system("epstopdf sqrt2sra.eps"); 
        system(    "open sqrt2sra.pdf"); //for LINUX 


//z=3.5;c=exq(z); printf("exq(3.5)=%19.4e %19.4e\n",Re(c),Im(c));
//z=3.;c=exq(z); printf("exq(3.0)=%19.4e %19.4e\n",Re(c),Im(c));
//      getchar(); system("killall Preview"); // For macintosh
}

Latex generator of the labels

\documentclass[12pt]{article} 
\usepackage{geometry} 
\usepackage{graphicx} 
\usepackage{rotating} 
\paperwidth 1230pt 
\paperheight 798pt 
\topmargin -100pt 
\oddsidemargin -75pt 
\textwidth 1640pt 
\textheight 1600pt 
\pagestyle {empty} 
\newcommand \sx {\scalebox} 
\newcommand \rot {\begin{rotate}} 
\newcommand \ero {\end{rotate}} 
\newcommand \ing {\includegraphics} 
\parindent 0pt
\pagestyle{empty} 
\begin{document} 

%\begin{picture}(360,740) 
\begin{picture}(1360,790) 
%\put(10,10){\ing{IterPowPlot}} 
%\put(40,40){\ing{Itereq2tlo}} 
%\put(0,0){\ing{sqrt26plo}} 
\put(0,0){\ing{sqrt2sra}} 
\put(590,767){\sx{4}{$y$}}
\put(590,697){\sx{4}{$6$}}
\put(590,597){\sx{4}{$5$}}
\put(590,497){\sx{4}{$4$}}
\put(590,396){\sx{4}{$3$}}
\put(590,296){\sx{4}{$2$}}
\put(590,196){\sx{4}{$1$}}
\put(590, 96){\sx{4}{$0$}}
\put(90,50){\sx{4}{$-5$}}
\put(180,50){\sx{4}{$-4$}}
\put(280,50){\sx{4}{$-3$}}
\put(380,50){\sx{4}{$-2$}}
\put(480,50){\sx{4}{$-1$}}
\put(612,50){\sx{4}{$0$}}
\put(712,50){\sx{4}{$1$}}
\put(812,50){\sx{4}{$2$}}
\put(912,50){\sx{4}{$3$}}
\put(1012,50){\sx{4}{$4$}}
\put(1112,50){\sx{4}{$5$}}
\put(1200,50){\sx{4}{$x$}}
\put(778,644){\sx{5}{$y\!=\!\exp_{\sqrt{2},4}^{~ 1/2}(x)$}}
\put(220,150){\sx{5}{$y\!=\!\exp_{\sqrt{2},2}^{~ 1/2}(x)$}}
\put(900,152){\sx{5}{$y\!=\!10^{24}D(x)$}}
\end{picture} 
\end{document} 

Refereces

Keywords

«Abelexponential», «Abelfunction», «Arctetration», «Base sqrt2», «Book», «BookPlot», «Explicit plot», «Exponential», «Iterate half», «Iterate», «Mathematics of Computation», «Sledgehammer to crack a nut», «Superexponential», «Superfunction», «Superfunctions»,

«Суперфункции»,