File:Sqrt2srav.png

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Comparison or two half iterate of exponent to base \( \sqrt{2} \) constructed at fixed point 2 and at fixed point 4.

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

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

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

Usage: this is figure 16.9 of the book Суперфункции (2014, In Russian) [1]; the English version is in preparation in 2015.

This image is used also in article [2].


Refereces

  1. ↑ https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
    http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
    http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.
  2. ↑ http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
    http://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf offprint D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.


C++ generator of the curves

Files ado.cin, conto.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}

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