File:Amosaplot.jpg

Explicit plot of amplitude of oscillator function, thick black curve, and its asymptotic approximation.

$y=\,$amos$(x)= \displaystyle \frac{e^{-x/2} \,\sqrt{x!}} {\pi^{1/4} (x/2)! }$

C++ generator of cirves
#include  //using namespace std; typedef std::complex z_type; //#include "facp.cin" //#include "afacc.cin"
 * 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 "fac.cin"

//z_type Amp(z_type n){ return sqrt(fac(n)/sqrt(M_PI))/( exp((log(2.)/2.)*n)*fac(.5*n));} z_type Amp(z_type n){ return exp( -.5*log(2.)*n + (.5*lof(n)-lof(.5*n)) )/sqrt(sqrt(M_PI));} z_type Ama(z_type n){ DB c[22]=  {1, -0.125, 0.0078125, 0.0205078125 , -0.0025939941406250, -0.02483749389648438, 0.003335237503051758, 0.07567062973976135, -0.009997612331062555, -0.4298963562468998, 0.05568409210172831, 3.922531476012864, -0.5014786647962097, -52.47756609951033, 6.657774463390167, 967.8787886035127, -122.2445554325703, -23538.53880986894, 2964.833600121925, 729848.1121739772,-91766.94357152004,-2.810206798285763e7}; z_type s; int m,M=21; s=c[M]; for(m=M-1;m>0;m--){s/=n; s+= c[m];} return (1.+s/n)*sqrt((sqrt(2.)/M_PI)/sqrt(n));}


 * 1) include "ado.cin"

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; for(n=0;n<11;n++) {x=Re(Amp(0.+n)); y=sqrt((sqrt(2.)/M_PI)/sqrt(n+.5)); //t=Re(Ama(0.+n)); printf("%2d %20.14lf %20.14lf %20.14lf\n",n,x,y,x-y);}

FILE *o;o=fopen("amosaplo.eps","w");ado(o,504,174); fprintf(o,"102 2 translate\n 100 100 scale 2 setlinecap 1 setlinejoin\n");
 * 1) define M(x,y) fprintf(o,"%6.4lf %6.4lf M\n",x+0.,y+0.);
 * 2) define L(x,y) fprintf(o,"%6.4lf %6.4lf L\n",x+0.,y+0.);

for(m=-1;m<5;m++){ M(m,0)L(m,1) } for(n=0;n<5;n++){ M( -1,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");

for(n=1;n<10;n++){ M( -1,.1*n)L(4,.1*n)} fprintf(o,".004 W 0 0 0 RGB S\n");

//DO(n,50){ x=-.82+.005*(n*n); y=Re(Amp(x)); if(n==0)M(x,y) else L(x,y);} DO(n,50){ x=-.86+.002*(n*(n+2)); y=Re(Amp(x)); if(n==0)M(x,y) else L(x,y);} fprintf(o,".015 W 0 0 0 RGB S\n");

/* DO(n,59){x=.14+.004*(n*n); y=sqrt((sqrt(2.)/M_PI)/sqrt(x)); if(n==0)M(x,y) else L(x,y); if(x>4) break;} fprintf(o,".008 W 0 0 1 RGB S\n"); DO(n,59){x=.16+.003*(n*n); y=sqrt((sqrt(2.)/M_PI)/sqrt(x))*(1.-1./8./x); if(n==0)M(x,y) else L(x,y); if(x>4) break;} fprintf(o,".008 W 1 0 0 RGB S\n"); DO(n,59){x=-.4+.002*(n*n); y=sqrt((sqrt(2.)/M_PI)/sqrt(x+.5)); if(n==0)M(x,y) else L(x,y); if(x>4) break;} fprintf(o,".008 W 0 .8 0 RGB S\n");

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

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \paperwidth 504pt \paperheight 188pt \topmargin -96pt \oddsidemargin -73pt \pagestyle{empty} \usepackage{graphicx} \usepackage{rotating} \parindent 0pt \textwidth 800px \textheight 900px \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document} \begin{picture}(506,160) \put(0,0){\includegraphics{amosaplo}} %\put(20,10){\includegraphics{amosma}} %\put(20,10){\includegraphics{lofma}} %\put(20,10){\includegraphics{hermiga6ma}} %\put(20,10){\includegraphics{hermiten6draft}} %\put(84,116){\sx{1.5}{$y$}} \put(84,96){\rot{0}\sx{1.5}{$1$}\ero} \put(77,46){\rot{0}\sx{1.5}{$0.5$}\ero} \put(84,-4){\rot{0}\sx{1.5}{$0$}\ero} \put(99,-14){\rot{0}\sx{1.5}{$0$}\ero} \put(199,-14){\rot{0}\sx{1.5}{$1$}\ero} \put(299,-14){\rot{0}\sx{1.5}{$2$}\ero} \put(399,-14){\rot{0}\sx{1.5}{$3$}\ero} \put(495,-14){\rot{0}\sx{1.5}{$x$}\ero} % \put(20,159){\sx{1.5}{$y\!=\!\mathrm{amos}(x)$}} %\put(50,122){\sx{1.5}{$y\!=\! (2/(x\!+\!1/2))^{1/4}\pi^{-1/2}$}} \put(54,124){\sx{1.3}{$\displaystyle y\!=\! \pi^{-1/2} \left( \frac{2}{x\!+\!1/2}\right)^{\!1/4}$}} %\put(104,21){\sx{1.4}{$y\!=\! (2/x)^{1/4}\pi^{-1/2}(1-\frac{1}{8x})$}} %\put(104,18.4){\sx{1.5}{$y\!=\! (2/x)^{1/4}\pi^{-1/2}(1-\frac{1}{8x})$}}

\end{picture} \end{document}