File:Noriplot300.jpg

Integrand for the transmittance of the Bessel mode through the slit for the teal argument; square of the Morinaga function mori of modified argument:

$y=\mathrm{nori}(x)=\mathrm{mori}\big(\sqrt{x}\big)^2$, blue curve, and

and

$y=100 \,\mathrm{nori}(x)=100 \, \mathrm{mori}\big(\sqrt{x}\big)^2$, red curve.

The additional grid line indicates the abscissa $x=L_2^2/L_1^2\approx 5.2689404316052215$ ,

where

$L_1=\mathrm{BesselJZero}[0,1]\approx 1.2484591696955065 $

$L_2=\mathrm{BesselJZero}[0,2]\approx 5.5200781102863115 $

C++ generator of curves
File ado.cin should be loaded //#include "scft.cin" //#define NP 16 DB L1= 2.404825557695773; DB L2= 5.5200781102863115; DB L3= 8.653727912911013; //DB mory(DB x){ return j0(x)/(1.-(1./(L1*L1))*x*x);} DB morin(DB x){ return j0(L1*x)/(1-x*x);} DB mori0(DB x){ int n,m; DB 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 or bug in the Mathematica routine Series applied to HankelH. //with m=15, at |x|<2, the 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;} DB mori(DB x){if(fabs(x)<2) return mori0(x); return morin(x);}
 * 1) include 
 * 2) include 
 * 3) include
 * 1) define DB double
 * 2) include "ado.cin"

int main{ int i; //double a[NP+1],b[NP+1]; double d=sqrt(M_PI/NP); //double x,y,f,g; FILE *o; o=fopen("11.eps","w"); ado(o,1020,130); double x,y,f,g; FILE *o; o=fopen("noriplo.eps","w"); ado(o,1020,130); fprintf(o,"10 10 translate 50 100 scale 2 setlinecap 1 setlinejoin\n"); for(i=0;i<41;i++){M(.5*i,0)L(.5*i,1)} for(i=0;i<3;i++){M(0,.5*i)L(20,.5*i)} fprintf(o,".006 W S\n"); x=L2/L1; x*=x; M(x,.5)L(x,-.02) x=L3/L1; x*=x; M(x,.5)L(x,-.02) fprintf(o,".004 W S\n"); //for(i=0;i<101;i++){x=.01*L1*i; y=j0(x); if(i==0)M(x,y) else L(x,y); } L(L1,0)L(618,0) fprintf(o,".009 W 1 0 0 RGB S\n"); //for(i=0;i<122;i++){x=.05*(i-.1);y=morin(x); if(i==0)M(x,y) else L(x,y);} fprintf(o,".009 W 0 0 1 RGB S\n"); //for(i=0;i<122;i++){x=.05*(i-.1);y=mori0(x); if(i==0)M(x,y) else L(x,y); if(fabs(y)>1) break;} fprintf(o,".009 W 0 .9 0 RGB S\n"); //for(i=0;i<122;i++){x=.05*i;y=mori(x); if(i==0) M(x,y) else L(x,y); if(fabs(y)>1) break;} fprintf(o,".009 W 0 0 1 RGB S\n"); for(i=0;i<402;i++){x=.1*i;y=mori(sqrt(x)); y*=y; if(i==0) M(x,y) else L(x,y);} fprintf(o,".009 W 0 0 1 RGB S\n"); for(i=0;i<332;i++){x=3.7+.1*i;y=mori(sqrt(x)); y*=y; y*=100; if(i==0) M(x,y) else L(x,y);} fprintf(o,".009 W 1 0 0 RGB S\n"); /* for(i=1;i<640;i++){ x=.01*(i-.5); f=mori0(x); g=morin(x); if(f==g){y=9.;} else { y=-log(fabs(f-g)/(fabs(f)+fabs(g))); y/=log(10.); y/=2.;}; if(i==1)M(x,y) else L(x,y); printf("%5.2lf %20.14lf %20.14lf %10.4lf\n",x,f,g,y);} fprintf(o,".008 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer\n",'%','%'); fclose(o); system("epstopdf noriplo.eps"); system(   "open noriplo.pdf"); }
 * 1) define M(x,y) fprintf(o,"%9.4lf %9.4lf M\n",x+0.,y+0.);
 * 2) define L(x,y) fprintf(o,"%9.4lf %9.4lf L\n",x+0.,y+0.);

Latex generator of curves
\documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphics} \usepackage{rotating} \paperwidth 570pt \paperheight 120pt \textwidth 420pt \textheight 300pt \topmargin -108pt \oddsidemargin -73pt \newcommand \ds {\displaystyle} \newcommand \sx {\scalebox} \newcommand \rme {\mathrm{e}} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \pagestyle{empty} \parindent 0pt \begin{document} \begin{picture}(810,116) \put(0,0){\ing{noriplo}} %\put(0,0){\ing{11}} %\put(.6,120){\sx{1.25}{$y\!=\!\exp(-x^2/2)$ and its discrete representation with CFT at $N\!= 4$}} %\put(.6,120){\sx{1.25}{$y$}} \put(.6,104){\sx{1.3}{$1$}} \put(.6,56){\sx{1.4}{$\frac{1}{2}$}} \put(.6, 5){\sx{1.3}{$0$}} \put(57,-2){\sx{1.3}{$1$}} \put(107,-2){\sx{1.3}{$2$}} \put(157,-2){\sx{1.3}{$3$}} \put(207,-2){\sx{1.3}{$4$}} \put(258,-2){\sx{1.3}{$5$}} \put(308,-2){\sx{1.3}{$6$}} \put(358,-2){\sx{1.3}{$7$}} \put(408,-2){\sx{1.3}{$8$}} \put(459,-2){\sx{1.3}{$9$}} \put(504,-2){\sx{1.3}{$10$}} \put(562,-1.6){\sx{1.3}{$x$}} %\put(120,90){\rot{-28}\sx{1.3}{$y\!=\! J_0(x)\theta(L_1\!-\!x)$}\ero} %\put(112,58){\rot{-33}\sx{1.3}{$y\!=\! \mathrm{mori}(x)$}\ero} \put(28,84){\sx{1.3}{$y\!=\! \mathrm{mori}\Big(\sqrt{x}\Big)^2$}} \put(206,89){\sx{1.3}{$y\!=\! 100\,\mathrm{mori}\Big(\sqrt{x}\Big)^2$}} \end{picture}\end{document}