File:Esqrt2ite12mapT80.jpg

Complex map of the 1/2 iterate of the exponential to base sqrt(2), which is $T(z)= \exp_{\sqrt{2}}(z)=\big(\sqrt{2}\big)^z$

the $\frac{1}{2}$ iteration is function $f$ such that $f(f(z))=T(z)$

$u+\mathrm i v = f(x+\mathrm i y)$

C++ generator of curves
// Files ado.cin and conto.cin should be loaded to the working directory in order to compile the C++ code below.

using namespace std; typedef complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) include "conto.cin"

z_type f45E(z_type z){int n; z_type e,s; DB coeu[21]={1., 0.44858743119526122890, .19037224679780675668, 0.77829576536968278770e-1, 0.30935860305707997953e-1, 0.12022125769065893274e-1, 0.45849888965617461424e-2, 0.17207423310577291102e-2, 0.63681090387985537364e-3, 0.23276960030302567773e-3, 0.84145511838119915857e-4, 0.30115646493706434120e-4, 0.10680745813035087964e-4, 0.37565713615564248453e-5, 0.13111367785052622794e-5, 0.45437916254218158081e-6, 0.15642984632975371803e-6, 0.53523276400816416929e-7, 0.18207786280204973113e-7, 0.61604764947389226744e-8, 0.2e-8}; e=exp(.32663425997828098238*(z-1.11520724513161)); s=coeu[20]; for(n=19;n>=0;n--) { s*=e; s+=coeu[n]; } //    s=coeu[19]; for(n=18;n>=0;n--) { s*=e; s+=coeu[n]; } return 4.+s*e;}

z_type F45E(z_type z){ DB b=sqrt(2.); if(Re(z)<-1.) return f45E(z); return exp(F45E(z-1.)*log(b)); }

z_type f45L(z_type z){ int n; z_type e,s; DB Uco[21]={1, -.44858743119526122890,       .21208912005491969757,  -.10218436750697167872,        0.49698683037371830337e-1, -0.2430759032611955221e-1,    0.11933088396510860210e-1, -0.587369764200886206e-2,     0.289686728710575713e-2, -0.1430908106079253664e-2,    0.7076637148565759223e-3, -0.3503296158729878e-3,       0.17357560046634138e-3, -0.86061011929162626e-4,      0.426959089012891e-4, -0.2119302906819844809e-4,    0.1052442259960e-4, -0.52285174354086e-5,         0.259844999161e-5, -0.129178211214818578e-5,     0.4e-6  }; z-=4.; s=Uco[19]; for(n=18; n>=0; n--){ s*=z; s+=Uco[n]; } //    s=Uco[20]; for(n=19; n>=0; n--){ s*=z; s+=Uco[n]; } //     return log(s*z)/.32663425997828098238 +1.1152091357215375; return log(s*z)/.32663425997828098238 +1.11520724513161; }

z_type F45L(z_type z){ DB b=sqrt(2.); if(abs(z-4.)>.4) return F45L(log(z)/log(b))+1. ;                        return f45L(z); } // #include"sqrt2f45E.cin"

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; int M=801,M1=M+1; int N=405,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("esqrt2ite12map.eps","w"); ado(o,202,202); fprintf(o,"101 101 translate\n 10 10 scale\n"); DO(m,M1) X[m]=-10.+.025*(m-.5); //DO(n,N1) Y[n]=-10.+.04*(n-.5); // DO(n,200) Y[n]=sinh(3.*(n-200.5)/200.); // DO(n,200) Y[n]=-10.+.05*(n-.5); //        Y[200]=-.0001; //        Y[201]= .0001; for(n=0;n-25. && p<25. && q>-25. && q<25. //             && fabs(p)>1.e-14 //              && fabs(q)>1.e-14                ) { g[m*N1+n]=p; f[m*N1+n]=q;} }} p=.4; q=.3; for(m=-10;m<10;m++)for(n=2             ;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);  fprintf(o,".014 W 0 .7 0 RGB S\n"); for(m=0;m<11;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);                 fprintf(o,".014 W 1 0 0 RGB S\n"); for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);                   fprintf(o,".014 W 0 0 1 RGB S\n"); for(m= 1;m<25;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".03 W .8 0 0 RGB S\n"); for(m= 1;m<25;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".03 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".03 W .5 0 .5 RGB S\n"); for(m=-24;m<25;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");

// #include "plofu.cin" M(-10,0)L(2,0)fprintf(o,"0 setlinecap .036 W 1 1 1 RGB S\n"); for(n=0;n<27;n++){ M(2-.5*(n+.2),0) L(2-.5*(n+.4),0) } fprintf(o,".06 W 1 .5 0 RGB S\n"); for(n=0;n<27;n++){ M(2-.5*(n+.7),0) L(2-.5*(n+.9),0) } fprintf(o,".06 W 0 .5 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf esqrt2ite12map.eps"); system(   "open esqrt2ite12map.pdf"); //for macintosh }

Latex generator of curves
\documentclass[12pt]{article} \paperwidth 2072px \paperheight 2090px \textwidth 2394px \textheight 2300px \topmargin -86px \oddsidemargin -78px \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{10}{\begin{picture}(206,206) \put(6,5){\ing{esqrt2ite12map}} \put(2,203.4){\sx{.7}{$y$}} \put(2,184){\sx{.6}{$8$}} \put(2,164){\sx{.6}{$6$}} \put(2,144){\sx{.6}{$4$}} \put(2,124){\sx{.6}{$2$}} \put(2,104){\sx{.6}{$0$}} \put(-3,84){\sx{.6}{$-2$}} \put(-3,64){\sx{.6}{$-4$}} \put(-3,44){\sx{.6}{$-6$}} \put(-3,24){\sx{.6}{$-8$}} \put(-2,-1){\sx{.7}{$-\!10$}} \put( 22,-1){\sx{.7}{$-8$}} \put( 42,-1){\sx{.7}{$-6$}} \put( 62,-1){\sx{.7}{$-4$}} \put( 82,-1){\sx{.7}{$-2$}} \put(106,-1){\sx{.7}{$0$}} \put(126,-1){\sx{.7}{$2$}} \put(146,-1){\sx{.7}{$4$}} \put(166,-1){\sx{.7}{$6$}} \put(186,-1){\sx{.7}{$8$}} \put(204,-1){\sx{.7}{$x$}} \put(056,103.5){\sx{.99}{\bf cut}} \put(179,135.6){\sx{.9}{\rot{-19}$v\!=\!6$\ero}} \put(175,126){\sx{.9}{\rot{-14}$v\!=\!4$\ero}} \put(173,116){\sx{.9}{\rot{-10}$v\!=\!2$\ero}} \put(172,103.2){\sx{.9}{$v\!=\!0$}} \put(172,091.7){\sx{.9}{\rot{6}$v\!=\!-2$\ero}} \put(174,080.4){\sx{.9}{\rot{12}$v\!=\!-4$\ero}}

\put(066,161){\sx{.9}{\rot{4}$u\!=\!-2$\ero}} \put(076,138.4){\sx{.9}{\rot{25}$u\!=\!-1$\ero}} \put(098,121){\sx{.9}{\rot{54}$u\!=\!0$\ero}} \put(116.4,114){\sx{.9}{\rot{68}$u\!=\!1$\ero}} \put(129.8,110){\sx{.9}{\rot{76}$u\!=\!2$\ero}} \put(149.6,105){\sx{.9}{\rot{81}$u\!=\!4$\ero}} \put(164.4,103){\sx{.9}{\rot{83}$u\!=\!6$\ero}} \end{picture}} \end{document}