File:Sqrt2q2map600.jpg

Complex map of the half iterate of exponent to base $\sqrt{2}$ regular at its lowest fixed point.

$u\!+\!\mathrm i v= \exp_{\sqrt{2},\mathrm d}^{~ 1/2}(x\!+\!\mathrm i y)$

This function is expressed through tetration and arctetration to base $\sqrt{2}$:

$\displaystyle \exp_{\sqrt{2},\mathrm d}^{~ 1/2}(z)= \mathrm{tet}_{\sqrt{2}}\left( \frac{1}{2}+\mathrm{ate}_{\sqrt{2}}(z) \right)$

Usage: this is figure 16.8 of the book Суперфункции (2014, In Russian) ; the English version is in preparation in 2015.

The same map appears also at the top left picture of figure 6 of article .

C++ generator of the map
Files ado.cin, conto.cin, and

sqrt2f21e.cin, sqrt2f21l.cin,

or

sqrt2f23e.cin, sqrt2f23l.cin

should be loaded in order to compile the code below.

typedef std::complex z_type;
 * 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.)
 * 4) include "conto.cin"

// #include "uq2e.cin" // #include "uq2L.cin" // #include "f43E.cin" // #include "f43L.cin" //#include "sqrt2f23e.cin" //#include "sqrt2f23l.cin"
 * 1) include "sqrt2f21e.cin"
 * 2) include "sqrt2f21l.cin"

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; int M=211,M1=M+1; int N=201,N1=N+1; DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array. char v[M1*N1]; // v is working array FILE *o;o=fopen("sqrt26a.eps","w"); ado(o,214,212); fprintf(o,"112 110 translate\n 10 10 scale\n"); DO(m,M1) X[m]=-11+.1*(m-.5); DO(n,N1) Y[n]=-10+.1*(n-.5); for(m=-10;m<11;m++) {M(m,-10)L(m,10)} for(n=-10;n<11;n++) {M( -10,n)L(10,n)} fprintf(o,".006 W 0 0 0 RGB S\n"); //fprintf(o,"/adobe-Roman findfont 1 scalefont setfont\n"); // for(m=-8;m<0;m+=2) {M(-11.2,m-.3) fprintf(o,"(%1d)s\n",m);} // for(m= 0;m<9;m+=2) {M(-10.7,m-.3) fprintf(o,"(%1d)s\n",m);} // for(m=-8;m<0;m+=4) {M(m-.6,-10.8) fprintf(o,"(%1d)s\n",m);} // for(m= 0;m<9;m+=4) {M(m-.3,-10.8) fprintf(o,"(%1d)s\n",m);}

// fprintf(o,"/Times-Italic findfont 1 scalefont setfont\n"); //fprintf(o,"/adobe-italic findfont 1 scalefont setfont\n"); // M(-10.7, 9.5) fprintf(o,"(y)s\n"); // M( 9.6,-10.8) fprintf(o,"(x)s\n"); // M(-11,0)L(10.1,0) M(0,-11)L(0,10.1) fprintf(o,".01 W 1 0 1 RGB S\n"); // z_type tm,tp,F[M1*N1];; DO(m,M1)DO(n,N1){     g[m*N1+n]=9999; f[m*N1+n]=9999;} DO(m,M1){x=X[m]; DO(n,N1){y=Y[n]; z=z_type(x,y); if(abs(z-10.)>2.9) { //    c=UQ2E(z) ; d=uq2LA(c); //    c=UQ2L(z); c=UQ2E(c+.5); //    c=F43L(z); c=F43E(c+.5); c=F23L(z); c=F23E(c+.5); //   p=abs((z-d)/(z+d)); p=-log(p)/log(10.); p=Re(c); q=Im(c); if(p>-99 && p<99 && fabs(p)>1.e-12) g[m*N1+n]=p; if(q>-99 && q<99 && fabs(q)>1.e-12) f[m*N1+n]=q; }}} p=2; q=1;
 * 1) include "plofu.cin"

// p=1.e-15; //for(n=-10;n<11;n++) {q=p*n; z=z_type(q,0.); printf("%19.15f %19.16f\n",q, Re(uq2e(z))  ); } //M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n");

/* y=9.064720284; M(.89, y)L(10.1, y) M(.89,-y)L(10.1,-y) M(4,0)L( 10,0)fprintf(o,".15 W 0 0 0 RGB [.2 .2] 0 setdash S\n"); //The cuts live in a separate file, but one will be more stressed: M(4,0)L( 10.1,0) fprintf(o,".1 W 1 1 1 RGB S\n"); DO(n,28){M(4+.3*n,0)L(4+.3*(n+.5),0)} fprintf(o,".13 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf sqrt26a.eps"); system(   "open sqrt26a.pdf"); // for LINUX //     getchar; system("killall Preview"); // For macintosh }

C++ generator of the cut lines
Files ado.cin, sqrt2f21e.cin, sqrt2f21l.cin should be loaded in order to compile the code below.

typedef std::complex z_type;
 * 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.)
 * 4) include "conto.cin"

// #include "uq2e.cin" // #include "uq2L.cin" // #include "f43E.cin" // #include "f43L.cin"

// #include "f23E.cin"
 * 1) include "sqrt2f21L.cin"

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; int M= 60,M1=M+1; int N=201,N1=N+1; DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array. char v[M1*N1]; // v is working array FILE *o;o=fopen("sqrt26cuts.eps","w"); ado(o,214,212); fprintf(o,"112 110 translate\n 10 10 scale\n"); DO(m,M1) X[m]= 4.2 +.1*(m-.5); DO(n,N1) Y[n]=-10+.1*(n-.5); for(m=-10;m<11;m++) {M(m,-10)L(m,10)} for(n=-10;n<11;n++) {M( -10,n)L(10,n)} fprintf(o,".006 W 0 0 0 RGB S\n"); // fprintf(o,"/adobe-Roman findfont 1 scalefont setfont\n"); // for(m=-8;m<0;m+=2) {M(-11.2,m-.3) fprintf(o,"(%1d)s\n",m);} // for(m= 0;m<9;m+=2) {M(-10.7,m-.3) fprintf(o,"(%1d)s\n",m);} // for(m=-8;m<0;m+=4) {M(m-.6,-10.8) fprintf(o,"(%1d)s\n",m);} // for(m= 0;m<9;m+=4) {M(m-.3,-10.8) fprintf(o,"(%1d)s\n",m);} // fprintf(o,"/Times-Italic findfont 1 scalefont setfont\n"); // //fprintf(o,"/adobe-italic findfont 1 scalefont setfont\n"); // M(-10.7, 9.5) fprintf(o,"(y)s\n"); // M( 9.6,-10.8) fprintf(o,"(x)s\n"); // M(-11,0)L(10.1,0) M(0,-11)L(0,10.1) fprintf(o,".01 W 1 0 1 RGB S\n"); // z_type tm,tp,F[M1*N1];; DO(m,M1)DO(n,N1){     g[m*N1+n]=9999; f[m*N1+n]=9999;} DO(m,M1){x=X[m]; DO(n,N1){y=Y[n]; if(fabs(fabs(y)-9.)>.3) {       z=z_type(x,y); c=F21L(z); p=Re(c); q=Im(c); if(p>-99 && p<99) g[m*N1+n]=p; if(q>-99 && q<99 && fabs(q)>1.e-12) f[m*N1+n]=q; }}} //#include "plofu.cin"

conto(o,f,w,v,X,Y,M,N, (0. ), -2,2); // fprintf(o,".05 W 0 .8 0 RGB S\n");

/* y=9.064720284; M(.89, y)L(10.1, y) M(.89,-y)L(10.1,-y) M(4,0)L( 10,0)

// fprintf(o,".1 W 0 0 0 RGB [.2 .2] 0 setdash S\n");

y=9.064720284; M(.89, y)L(10.1, y) M(.89,-y)L(10.1,-y) M(4,0)L( 10,0)fprintf(o,".07 W 0 0 0 RGB [.2 .2] 0 setdash S\n");

fprintf(o,"[99]0 setdash\n");

/* M(2,0)L(-11.2,0)fprintf(o,".1 W 1 1 1 RGB S\n"); DO(n,36){M(2-.3*n,0)L(2-.3*(n+.5),0)} fprintf(o,".15 W 0 0 0 RGB S\n"); //M(2,0)L(-11,0)fprintf(o,".15 W 0 0 0 RGB [.2 .2] 0 setdash S\n"); //may cause problems

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf sqrt26cuts.eps"); system(   "open sqrt26cuts.pdf"); //     getchar; system("killall Preview"); // For macintosh }

Latex combiner
\documentclass[12pt]{article} \paperwidth 212px \paperheight 210px \textwidth 1394px \textheight 1300px \topmargin -100px \oddsidemargin -74px \usepackage{graphics} \usepackage{rotating} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \parindent 0pt \pagestyle{empty} \begin{document}\parindent 0pt

\sx{1.}{\begin{picture}(280,203) %\put(0,0){\ing{exc2cuts}} %\put(0,0){\ing{figexc2}} \put(0,0){\ing{sqrt26cuts}} \put(0,0){\ing{sqrt26a}} \put(6,207.7){\sx{.7}{$y$}} \put(6,187.8){\sx{.7}{$8$}} \put(6,167.8){\sx{.7}{$6$}} \put(6,147.7){\sx{.7}{$4$}} \put(6,127.7){\sx{.7}{$2$}} \put(6,107.7){\sx{.7}{$0$}} \put(0, 87.7){\sx{.7}{$-2$}} \put(0, 67.6){\sx{.7}{$-4$}} \put(0, 47.6){\sx{.7}{$-6$}} \put(0, 27.5){\sx{.7}{$-8$}} \put(3, 2){\sx{.7}{$-10$}} \put( 27,2){\sx{.7}{$-8$}} \put( 47,2){\sx{.7}{$-6$}} \put( 67,2){\sx{.7}{$-4$}} \put( 87,2){\sx{.7}{$-2$}} \put(111,2){\sx{.7}{$0$}} \put(131,2){\sx{.7}{$2$}} \put(151,2){\sx{.7}{$4$}} \put(171,2){\sx{.7}{$6$}} \put(191,2){\sx{.7}{$8$}} \put(210,2){\sx{.7}{$x$}} %\put( 27,127){\sx{3}{$\exp_{b,2}^{[1/2]}$}} \put(23,198.6){\sx{.9}{$v\!=\!0$}} \put(33.7,92){\rot{90}\sx{.9}{$u\!=\!-1.2$}\ero} \put(59.6,96){\rot{90}\sx{.9}{$u\!=\!-1$}\ero} \put(102,98){\rot{90}\sx{.9}{$u\!=\!0$}\ero} \put(122,99){\rot{90}\sx{.9}{$u\!=\!1$}\ero} \put(136,99){\rot{90}\sx{.9}{$u\!=\!2$}\ero} %\put(215,207){\sx{.8}{\bf cut}} \put(195,199){\sx{.8}{\bf cut}} \put(197,184){\rot{24}{\sx{.78}{\bf cut}}\ero} \put(204.4,174.4){\rot{40}{\sx{.78}{\bf cut}}\ero} %\put(215,184){\sx{.8}{\bf cut}} %\put(215,175){\sx{.8}{\bf cut}} \put(195,108){\sx{.78}{\bf cut}} % \put(87,145){\rot{-44}\sx{.9}{$v\!=\!1$}\ero} \put(61,122){\rot{-15}\sx{.8}{$v\!=\!0.2$}\ero} \put(63,108){\sx{.9}{$v\!=\!0$}} \put(62, 94){\rot{13}\sx{.8}{$v\!=\!-0.2$}\ero} %\put(215, 42){\sx{.8}{\bf cut}} %\put(215, 33){\sx{.8}{\bf cut}} %\put(215, 25){\sx{.8}{\bf cut}} \put(196,32){\rot{-22}{\sx{.78}{\bf cut}}\ero} \put(195, 17){\sx{.78}{\bf cut}} %\put(-23, 17){\sx{1.}{$q\!=0$}} %\put(215, 10){\sx{.8}{\bf cut}} \put(23,17){\sx{.9}{$v\!=\!0$}} \end{picture}}

\end{document}