File:Amoscplot.jpg

Explicit plot of function Amos that expresses the amplitude of oscillator function at zero versus its number. However, here, the number of oscillator function is treated as argument $x$; and, in general, this argument can take not only non–negative integer values, but also other real values (and even complex values).

The central black curve shows

$y=\,$Amos$(x)$ $ =\pi^{-1/4} \exp\!\Big( \frac{1}{2}$Lof$(x)-\,$Lof$\big(\frac{x}{2}\big)-\frac{x}{2}\ln(2)\Big)$ $ =\displaystyle \frac {2^{x/2} \, \sqrt{x!} } {\pi^{1/4}\, (x/2)! } $

The upper blue curve shows the upper asymptotic approximation,

$\displaystyle y\!=\! (2/\pi)^{1/2}\, \big(2x\!+\!1\big)^{\!-1/4}$

The lower red curve shows the lower asymptotic approximation,

$ y\!=\! (2/\pi)^{1/2}\, \Big((2x\!+\!1)\big(1+ \frac{1/2}{(2x+1)^2}\big)\Big)^{\!-1/4}$

The effective big parameter of expansion $2x\!+\!1$ has sense of the energy of the corresponding state of harmonic oscillator, at least for integer $x$ that can be interpreted as number of this state.

C++ generator of cirves
Files ado.cin and fac.cin should be loaded in order to compile the code below //using namespace std; 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++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)

//#include "facp.cin" //#include "afacc.cin"
 * 1) 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));}


 * 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<21;n++) {x=Re(Amp(0.+n)); y=sqrt( (2./M_PI) / ( sqrt( 2.*n+1. ) )   ); t=sqrt((2./M_PI)/sqrt((2.*n+1.) * (1+1./2./((2.*n+1.)*(2.*n+1.))) )) ; printf("%2d %20.14lf %20.14lf %20.14lf %20.14lf %20.14lf\n",n,x,y,t,y-x,t-x);}

FILE *o;o=fopen("amoscplo.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,59){x=-.44+.002*(n*n); y=sqrt((2./M_PI)/sqrt(2*x+1.)); if(n==0)M(x,y) else L(x,y); if(x>4) break;} fprintf(o,".01 W 0 0 1 RGB S\n");

DO(n,59){x=-.495+.002*(n*n); y=sqrt((2./M_PI)/sqrt( (2*x+1.) * (1+1./2./((2*x+1.)*(2*x+1.)) )) ) ; if(n==0)M(x,y) else L(x,y); if(x>4) break;} fprintf(o,".01 W 1 0 0 RGB S\n");

DO(n,50){ x=-.88+.002*(n*(n+2)); y=Re(Amp(x)); if(n==0)M(x,y) else L(x,y);} fprintf(o,".01 W 0 0 0 RGB S\n");

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

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{amoscplo}} %\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(85,97){\rot{0}\sx{1.4}{$1$}\ero} \put(78,47){\rot{0}\sx{1.4}{$0.5$}\ero} \put(84,-4){\rot{0}\sx{1.4}{$0$}\ero} \put(99,-13){\rot{0}\sx{1.4}{$0$}\ero} \put(199,-13){\rot{0}\sx{1.4}{$1$}\ero} \put(299,-13){\rot{0}\sx{1.4}{$2$}\ero} \put(399,-13){\rot{0}\sx{1.4}{$3$}\ero} \put(495,-13){\rot{0}\sx{1.4}{$x$}\ero} % \put(20,159){\sx{1.4}{$y\!=\!\mathrm{amos}(x)$}} %\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(60,132){\sx{1}{$\displaystyle y\!=\! (2/\pi)^{1/2}\, \big(2x\!+\!1\big)^{\!-1/4}$}} %\put(60,132){\sx{1}{$ y\!=\! \left(\frac 2 \pi \right)^{\!1/2} \big(2x\!+\!1\big)^{\!-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})$}} \put(8,25){\sx{1}{$ y\!=\! (2/\pi)^{1/2}\, \Big((2x\!+\!1)\big(1+ \frac{1/2}{(2x+1)^2}\big)\Big)^{\!-1/4}$}} \end{picture} \end{document}