File:Norimap40.jpg

complex map of function nori, in some range of the complex plane,

$\mathrm{nori}(z)=\,$mori$\big(\sqrt{z}\big)^2$

$u\!+\!\mathrm i v=\mathrm{nori}(x\!+\!\mathrm i y)$

C++ generator of curves
Files ado.cin, conto.cin, besselj0.cin should be loaded in order to compile the code below: //using namespace std; typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) define DB double
 * 4) 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) define I z_type(0.,1.)
 * 4) include "conto.cin"
 * 5) include "besselj0.cin"

DB L1= 2.404825557695773; DB L2= 5.5200781102863115; DB L3= 8.653727912911013; DB L4=11.791534439014281; z_type morin(z_type x){ return BesselJ0(L1*x)/(1.-x*x);} // naive representation fails at x=1.

z_type mori0(z_type x){ int n,m; z_type s, xx=x*x; DB c[16]={ 1., -0.4457964907366961303, 0.07678538241994023453, -0.0071642885058902232688, 0.00042159522055140947688, -0.000017110542281627483109, 5.0832583976057607495e-7, -1.1537378620148452816e-8, 2.0662789231930073316e-10, -2.9948657413756059965e-12, 3.5852738451127332173e-14,-3.6050239634659700777e-16, 3.0877184831292878827e-18, -2.2798156440952688462e-20, 1.4660907878585489441e-22,-8.2852774398657968065e-25}; // 16th term seems to fail; perhaps, due to the C++ rounding errors. //with m=15, at |x|<2, the relative error is of order of 10^(-16) //In this sense, the result is accurate while |x|<2. m=15; s=c[m]*xx; for(n=m-1;n>0;n--){ s+=c[n]; s*=xx;} return 1.+s;} z_type mori(z_type x){if(abs(x)<2.) return mori0(x); return morin(x);}

z_type no1(z_type x){ return BesselJ0(L1*sqrt(x))/(1.-x);} // naive representation fails at x=1.

z_type no0(z_type x){ int n,m; z_type s; //, xx=x*x; DB c[16]={ 1., -0.4457964907366961303, 0.07678538241994023453, -0.0071642885058902232688, 0.00042159522055140947688, -0.000017110542281627483109, 5.0832583976057607495e-7, -1.1537378620148452816e-8, 2.0662789231930073316e-10, -2.9948657413756059965e-12, 3.5852738451127332173e-14,-3.6050239634659700777e-16, 3.0877184831292878827e-18, -2.2798156440952688462e-20, 1.4660907878585489441e-22,-8.2852774398657968065e-25}; // 16th term seems to fail; perhaps, due to the C++ rounding errors. //with m=15, at |x|<2, the relative error is of order of 10^(-16) //In this sense, the result is accurate while |x|<2. m=15; s=c[m]*x; for(n=m-1;n>0;n--){ s+=c[n]; s*=x;} return 1.+s;}

z_type no(z_type x){if(abs(x)<3.) return no0(x); return no1(x);}

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=501,M1=M+1; int N=501,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("06.eps","w");ado(o,5020,5020); FILE *o;o=fopen("norima.eps","w");ado(o,5020,5020); fprintf(o,"1010 1010 translate\n 100 100 scale\n"); DO(m,M1)X[m]=-10.+.1*(m-.5); DO(n,N1)Y[n]=-10.+.1*(n-.5);

for(m=-10;m<11;m++){if(m==0){M(m,-10.1)L(m,10.1)} else{M(m,-10)L(m,10)}} for(n=-10;n<11;n++){    M(  -10,n)L(10,n)} fprintf(o,".01 W 0 0 0 RGB S\n"); DO(m,M1)DO(n,N1){g[m*N1+n]=99999; f[m*N1+n]=99999;} DO(m,M1){x=X[m]; //printf("%5.2f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); //     c=BesselJ0(z); //     c=mori(z); c=no(z); c*=c; p=Re(c);   q=Im(c); if(p>-999. && p<999. &&     q>-999. && q<999      ) {g[m*N1+n]=p; f[m*N1+n]=q; }                    }} //#include "plodi.cin" fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=40;q=1; for(m=-4;m<4;m++)for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".01 W 0 .6 0 RGB S\n"); for(m=0;m<4;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".01 W .9 0 0 RGB S\n"); for(m=0;m<4;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".01 W 0 0 .9 RGB S\n"); for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".04 W .9 0 0 RGB S\n"); for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".04 W .6 0 .6 RGB S\n"); for(m=-8;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n"); m=0;         conto(o,g,w,v,X,Y,M,N, (0.+m),-2*p,2*p); fprintf(o,".04 W 0 0 0 RGB S\n"); for(m=1;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n"); //#include "plofu.cin" fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf norima.eps"); system(   "open norima.pdf"); }

Latex generator of labels
\documentclass[12pt]{article} \paperheight 5120px \paperwidth 5100px \textwidth 5094px \textheight 5200px \topmargin -80px \oddsidemargin -72px \usepackage{graphics} \usepackage{rotating} %\usepackage{color} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \begin{document}\parindent 0pt \begin{picture}(5100,5100) \put(80,80){\ing{norima}} %\put(0,0){\sx{8}{-10}} \put(32,2068){\sx{8}{$y$}} \put(32,1868){\sx{8}{$8$}} \put(32,1668){\sx{8}{$6$}} \put(32,1468){\sx{8}{$4$}} \put(32,1268){\sx{8}{$2$}} \put(32,1068){\sx{8}{$0$}} \put(-32,868){\sx{8}{$-2$}} \put(-32,668){\sx{8}{$-4$}} \put(-32,468){\sx{8}{$-6$}} \put(-32,268){\sx{8}{$-8$}} \put(230,8){\sx{8}{$-8$}} \put(430,8){\sx{8}{$-6$}} \put(630,8){\sx{8}{$-4$}} \put(830,8){\sx{8}{$-2$}} \put(1070,8){\sx{8}{$0$}} \put(1270,8){\sx{8}{$2$}} \put(1470,8){\sx{8}{$4$}} \put(1670,8){\sx{8}{$6$}} \put(1870,8){\sx{8}{$8$}} \put(2070,8){\sx{8}{$x$}} % \put(120,1100){\rot{2}\sx{8}{$v\!=\!-8$}\ero} \put(160,1032){\rot{-2}\sx{8}{$v\!=\!8$}\ero} \put(120,800){\rot{8}\sx{8}{$u\!=\!8$}\ero} \put(120,732){\rot{3}\sx{8}{$u\!=\!-8$}\ero} % \put(840,910){\rot{50}\sx{8}{$u\!=\!4$}\ero} \put(890,862){\rot{6}\sx{8}{$u\!=\!0$}\ero} \put(760,830){\rot{-32}\sx{8}{$u\!=\!-4$}\ero} % \put(1340,1070){\rot{0}\sx{8}{$v\!=\!0$}\ero} \put(1906,1070){\rot{0}\sx{8}{$v\!=\!0$}\ero} % \put(1450,840){\rot{38}\sx{8}{$v\!=\!0$}\ero} \put(1560,740){\rot{50}\sx{8}{$u\!=\!0$}\ero} \put(1680,610){\rot{50}\sx{8}{$v\!=\!0$}\ero} \put(1844,480){\rot{51}\sx{8}{$u\!=\!0$}\ero} \put(1980,290){\rot{50}\sx{8}{$v\!=\!0$}\ero} \end{picture} \end{document}