Difference between revisions of "File:MoriplotFragment.jpg"

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Fragment from image
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http://mizugadro.mydns.jp/t/index.php/File:Moriplot300.jpg
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The principal Bessel mode
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$y\!=\!J_0(x)\theta(x)$ , red line,
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and its scaled [[Bessel transform]], expressed through the [[Morinaga function]]
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$y\!=\,$[[mori]]$(x)\!=\!$ $\displaystyle \frac{J_0(L_1 x)}{1\!-\!x^2}$ , blue line.
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Here, $L_n$ is the $n$th zero of the Bessel function, $L_n\!=\,$[[BesselJZero]]$[0,n]$
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In particular, $L_1\!\approx\! 2.404825557695773$
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The presentation of the Bessel transform of the Bessel mode is expressed as follows:
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$\displaystyle
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\int_0^\infty \mathrm{mode}(x) \, J_0( p x)\, x\, \mathrm d x =\,$
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$\displaystyle
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\int_0^{L_1} J_0(x) \, J_0( p x)\, x\, \mathrm d x =\,$
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$\displaystyle
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L_1\, J_1(L_1) \frac{J_0(p)}{1\!-\!p^2} =\,$
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$\displaystyle
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J_1(L_1)\, L_1\, \mathrm{mori}(p)$
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==[[C++]] generator of curves==
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File [[ado.cin]] should be loaded in order to compile the code below.
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<poem><nomathjax><nowiki>
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#include <stdio.h>
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#include <math.h>
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#include<stdlib.h>
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//#include "scft.cin"
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#define DB double
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#include "ado.cin"
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DB L1= 2.404825557695773;
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DB L2= 5.5200781102863115;
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DB L3= 8.653727912911013;
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DB L4=11.791534439014281;
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//DB mory(DB x){ return j0(x)/(1.-(1./(L1*L1))*x*x);} // mory also could be principal, but for graphics mori is better.
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DB morin(DB x){ return j0(L1*x)/(1-x*x);} // naive representation fails at x=1.
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DB mori0(DB x){ int n,m; DB s, xx=x*x;
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DB c[16]={ 1., -0.4457964907366961303, 0.07678538241994023453, -0.0071642885058902232688,
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0.00042159522055140947688, -0.000017110542281627483109, 5.0832583976057607495e-7, -1.1537378620148452816e-8,
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2.0662789231930073316e-10, -2.9948657413756059965e-12, 3.5852738451127332173e-14,-3.6050239634659700777e-16,
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3.0877184831292878827e-18, -2.2798156440952688462e-20, 1.4660907878585489441e-22,-8.2852774398657968065e-25};
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// 16th term seems to fail; perhaps, due to the C++ rounding errors.
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//with m=15, at |x|<2, the relative error is of order of 10^(-16)
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//In this sense, the result is accurate while |x|<2.
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m=15; s=c[m]*xx; for(n=m-1;n>0;n--){ s+=c[n]; s*=xx;}
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return 1.+s;}
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DB mori(DB x){if(fabs(x)<2.) return mori0(x);
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return morin(x);}
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int main(){ int i; //double a[NP+1],b[NP+1]; double d=sqrt(M_PI/NP);
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//double x,y,f,g; FILE *o; o=fopen("08.eps","w"); ado(o,620,820);
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double x,y,f,g; FILE *o; o=fopen("moriplo.eps","w"); ado(o,620,120);
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#define M(x,y) fprintf(o,"%9.4lf %9.4lf M\n",x+0.,y+0.);
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#define L(x,y) fprintf(o,"%9.4lf %9.4lf L\n",x+0.,y+0.);
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fprintf(o,"10 10 translate 100 100 scale 2 setlinecap 1 setlinejoin\n");
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for(i=0;i<12;i++){M(.5*i,0)L(.5*i,1)}
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for(i=0;i<3;i++){M(0,.5*i)L(5.5,.5*i)} fprintf(o,".006 W S\n");
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M(L2/L1,1)L(L2/L1,-.04)
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M(L1,1)L(L1,-.04)
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M(L3/L1,1)L(L3/L1,-.04)
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M(L4/L1,1)L(L4/L1,-.04)
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fprintf(o,".003 W S\n");
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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");
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//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");
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//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");
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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");
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/*
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for(i=1;i<640;i++){ x=.01*(i-.5); f=mori0(x); g=morin(x); if(f==g){y=9.;}
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else { y=-log(fabs(f-g)/(fabs(f)+fabs(g))); y/=log(10.); y/=2.;};
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if(i==1)M(x,y) else L(x,y); printf("%5.2lf %20.14lf %20.14lf %10.4lf\n",x,f,g,y);}
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fprintf(o,".008 W 0 0 0 RGB S\n"); // this is agreement of functions morin and mori0; important at the testing.
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*/
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fprintf(o,"showpage\n%c%cTrailer\n",'%','%'); fclose(o);
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system("epstopdf moriplo.eps");
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system( "open moriplo.pdf");
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}
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</nowiki></nomathjax></poem>
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==[[Latex]] generator of labels==
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<poem><nomathjax><nowiki>
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</nowiki></nomathjax></poem>
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==Referencs==
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<references/>
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[[Category:Bessel function]]
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[[Category:Bessel transform]]
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[[Category:Explicit plot]]
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[[Category:Fragment]]
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[[Category:Makoto Morinaga]]
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[[Category:Morinaga function]]

Latest revision as of 08:43, 1 December 2018

Fragment from image http://mizugadro.mydns.jp/t/index.php/File:Moriplot300.jpg

The principal Bessel mode

$y\!=\!J_0(x)\theta(x)$ , red line,

and its scaled Bessel transform, expressed through the Morinaga function

$y\!=\,$mori$(x)\!=\!$ $\displaystyle \frac{J_0(L_1 x)}{1\!-\!x^2}$ , blue line.

Here, $L_n$ is the $n$th zero of the Bessel function, $L_n\!=\,$BesselJZero$[0,n]$

In particular, $L_1\!\approx\! 2.404825557695773$

The presentation of the Bessel transform of the Bessel mode is expressed as follows:

$\displaystyle \int_0^\infty \mathrm{mode}(x) \, J_0( p x)\, x\, \mathrm d x =\,$ $\displaystyle \int_0^{L_1} J_0(x) \, J_0( p x)\, x\, \mathrm d x =\,$ $\displaystyle L_1\, J_1(L_1) \frac{J_0(p)}{1\!-\!p^2} =\,$ $\displaystyle J_1(L_1)\, L_1\, \mathrm{mori}(p)$

C++ generator of curves

File ado.cin should be loaded in order to compile the code below.


#include <stdio.h>
#include <math.h>
#include<stdlib.h>
//#include "scft.cin"
#define DB double
#include "ado.cin"
DB L1= 2.404825557695773;
DB L2= 5.5200781102863115;
DB L3= 8.653727912911013;
DB L4=11.791534439014281;
//DB mory(DB x){ return j0(x)/(1.-(1./(L1*L1))*x*x);} // mory also could be principal, but for graphics mori is better.
DB morin(DB x){ return j0(L1*x)/(1-x*x);} // naive representation fails at x=1.
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.
//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;}
DB mori(DB x){if(fabs(x)<2.) return mori0(x);
                        return morin(x);}

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("08.eps","w"); ado(o,620,820);
double x,y,f,g; FILE *o; o=fopen("moriplo.eps","w"); ado(o,620,120);
#define M(x,y) fprintf(o,"%9.4lf %9.4lf M\n",x+0.,y+0.);
#define L(x,y) fprintf(o,"%9.4lf %9.4lf L\n",x+0.,y+0.);
fprintf(o,"10 10 translate 100 100 scale 2 setlinecap 1 setlinejoin\n");
for(i=0;i<12;i++){M(.5*i,0)L(.5*i,1)}
for(i=0;i<3;i++){M(0,.5*i)L(5.5,.5*i)} fprintf(o,".006 W S\n");
M(L2/L1,1)L(L2/L1,-.04)
M(L1,1)L(L1,-.04)
M(L3/L1,1)L(L3/L1,-.04)
M(L4/L1,1)L(L4/L1,-.04)
fprintf(o,".003 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=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"); // this is agreement of functions morin and mori0; important at the testing.
*/
fprintf(o,"showpage\n%c%cTrailer\n",'%','%'); fclose(o);
system("epstopdf moriplo.eps");
system( "open moriplo.pdf");
}

Latex generator of labels

Referencs

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