File:Sqrt2srav.png

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, $y=D(x)= \exp_{\sqrt{2},\mathrm u}^{~1/2}(x) -\exp_{\sqrt{2},\mathrm d}^{~1/2}(x)$ , svaled with factor $10^{24}$.

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

This image is used also in article .

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.

typedef std::complex z_type; // #include "conto.cin" // #include "superex.cin" // #include "slog14128.cin"
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 1) include "ado.cin"
 * 2) include "sqrt2f23E.cin"
 * 3) include "sqrt2f23L.cin"
 * 4) include "sqrt2f43E.cin"
 * 5) include "sqrt2f43L.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 o(x,y) fprintf(o,"%7.4f %7.4f o\n",0.+x,0.+y);
 * 1) define M(x,y) fprintf(o,"%7.4f %7.4f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%7.4f %7.4f L\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}