File:Hermiten6map.jpg

Complex map of the normalised Hermite polynomial number 6:

$u\!+\!\mathrm i v= h_6(x\!+\!\mathrm i y)$

in the $x$, $y$ plane;

$\displaystyle h_n(z)= \frac{ H_n(z)} {\sqrt{N_n}}= \frac{ \mathrm{HermiteH}[n,z]} {\sqrt{N_n}}$

$\displaystyle N_n=\int_{-\infty}^{\infty} H_n(x)^2 \exp(-x^2)\, \mathrm dx = 2^n\, n! \, \sqrt{\pi}$

C++ generator of curves
Files hermiteneve.txt, hermitenodd.txt, ado.cin, conto.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;x81) {printf("hermite number %2d is not yet implemented (max. is 81)\n consider to stop..",n); getchar; return 0;} xx=x*x; //printf("Herniten called with n=%3d x=%8.2lf\n",n,x); if(n/2*2==n){M=n/2; s=0.;for(m=M;m>0;m--) {s+=HermiteH0[M][m]; s*=xx;} return (HermiteH0[M][0]+s);} else{ M=(n-1)/2; s=0.; for(m=M;m>0;m--) {s+=HermiteH1[M][m]; s*=xx;} return (HermiteH1[M][0]+s)*x;} }

int main{ int j,k,m,n; DB x,y, p,q, t,r; z_type z,c,d; int M=301,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("hermiten6ma.eps","w");ado(o,604,404); fprintf(o,"302 202 translate\n 100 100 scale 2 setlinecap\n"); DO(m,M1) {X[m]=-3.+.02*(m-.5);} DO(n,N1) {Y[n]=-2.+.02*(n-.5);} for(m=-3;m<4;m++){M(m,-2)L(m,2)} for(n=-2;n<3;n++){M( -3,n)L(3,n)} fprintf(o,".008 W 0 0 0 RGB S\n"); DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;} DO(m,M1){x=X[m]; //printf("%5.2f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); c=hermiten(6,z); p=Re(c);q=Im(c); if(p>-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p; f[m*N1+n]=q;} }} fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=300.;q=3.; for(m=-4;m<4;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".005 W 0 .6 0 RGB S\n"); for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".005 W .9 0 0 RGB S\n"); for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".005 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,".012 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,".012 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".012 W .6 0 .6 RGB S\n"); for(m=-8;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".012 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf hermiten6ma.eps"); system(   "open hermiten6ma.pdf"); }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \paperwidth 630pt \paperheight 424pt \topmargin -104pt \oddsidemargin -68pt \pagestyle{empty} \usepackage{graphicx} \usepackage{rotating} \parindent 0pt \textwidth 700px \textheight 900px \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document} \begin{picture}(614,410) \put(20,10){\includegraphics{hermiten6ma}} %\put(20,10){\includegraphics{hermiten6draft}} \put(4,404){\sx{2}{$y$}} \put(4,305){\sx{2}{$1$}} \put(4,205){\sx{2}{$0$}} \put(-10,105){\sx{2}{$-1$}} \put(-11,6){\sx{2}{$-2$}} \put( 0,-8){\sx{2}{$-3$}} \put(100,-8){\sx{2}{$-2$}} \put(200,-8){\sx{2}{$-1$}} \put(318,-8){\sx{2}{$0$}} \put(418,-8){\sx{2}{$1$}} \put(518,-8){\sx{2}{$2$}} \put(612,-8){\sx{2}{$x$}}

\put(231,192){\sx{1.5}{\rot{90}$v\!=\!0$\ero}} \put(284,190){\sx{1.5}{\rot{90}$u\!=\!0$\ero}} \put(328,218){\sx{1.5}{\rot{90}$v\!=\!0$\ero}} \put(310,208){\sx{1.5}{\rot{0}$v\!=\!0$\ero}} \put(372,190){\sx{1.5}{\rot{90}$u\!=\!0$\ero}} \put(394,186){\sx{1.5}{\rot{89}$u\!=\!0.4$\ero}} \put(425,194){\sx{1.5}{\rot{90}$v\!=\!0$\ero}} \put(463,194){\sx{1.5}{\rot{90}$u\!=\!0$\ero}} % \put(394,146){\sx{1.5}{\rot{59}$u\!=\!1$\ero}} \put(400,134){\sx{1.5}{\rot{40}$u\!=\!2$\ero}}

\put(238,154){\sx{1.5}{\rot{24}$v\!=\!1$\ero}}% \put(238,136){\sx{1.5}{\rot{19}$v\!=\!2$\ero}} \put(238,124){\sx{1.5}{\rot{14}$v\!=\!3$\ero}}%

\put(284,138){\sx{1.5}{\rot{37}$u\!=\!-1$\ero}}%% \put(284,122){\sx{1.5}{\rot{25}$u\!=\!-2$\ero}} \put(286,112){\sx{1.5}{\rot{20}$u\!=\!-3$\ero}}%% % \put(337,138){\sx{1.5}{\rot{39}$v\!=\!-1$\ero}} \put(346,126){\sx{1.5}{\rot{30}$v\!=\!-2$\ero}} \put(346,112){\sx{1.5}{\rot{26}$v\!=\!-3$\ero}}

\put(580,316){\sx{1.5}{\rot{34}$v\!=\!-8$\ero}} \put(595,301){\sx{1.5}{\rot{37}$v\!=\!8$\ero}} % \put(580,258){\sx{1.5}{\rot{18}$u\!=\!-8$\ero}} \put(598,229){\sx{1.5}{\rot{35}$u\!=\!8$\ero}}

\end{picture} \end{document}