File:Besselj1mapT080.png: Difference between revisions

Summary

Title / Description	Complex map of the Bessel function, ${\displaystyle u+\mathrm {i} v=J_{0}(x+\mathrm {i} y)}$j_
Citizendium author & Copyright holder	Copyright © Dmitrii Kouznetsov. See below for licence/re-use information.
Date created	2012 April 29
Country of first publication	Japan
Notes	Copy from [1] </ref>
Other versions	http://tori.ils.uec.ac.jp/TORI/index.php/File:Besselj0map1T080.png
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Generators

besselj0.cin

The complex(double) C++ implementation besselj0.cin is used:

// The code below should be stored in the working directory as besselj0.cin

z_type BesselJ0o(z_type z){ int n; z_type c,s,t;
s=1.; c=1.; t=-z*z/4.; for(n=1;n<32;n++) {c/=0.+n*n; c*=t; s+=c;}
return s;}

z_type BesselJ0B(z_type z){ int n; z_type c,C,s,S,t,u,x;
t=M_PI/4.-z; c=cos(t); s=sin(t); u=1./16./(z*z);
C=(((((((((((
+   11021897833929133607268351617203125./137438953472.)*u
-       502860269940467106811189921875./8589934592.)*u
+          57673297952355815927071875./1073741824.)*u
-             1070401384414690453125./16777216.)*u 
+                213786613951685775./2097152.)*u
-                   30241281245175./131072.)*u 
+                     13043905875./16384.)*u 
-                        2401245./512.)*u 
+                          3675./64.)* u 
-                            9./4.)*u + 2.)* c;
S=(((((((((((
-   882276678992136837800861860405640625./274877906944.)*u
+      36232405765710498380237842265625./17179869184.)*u
-         3694483615889146090857721875./2147483648.)*u
+             60013837619516978071875./33554432.)*u 
-               10278202593831046875./4194304.)*u
+                  1212400457192925./262144.)*u
-                     418854310875./32768.)*u
+                        57972915./1024.)*u
-                          59535./128.)*u 
+                            75./8.)*u - 1.) *s/4./z;
return (C+S)/sqrt(2.*M_PI*z);}

z_type BesselJ0(z_type z){ if(Re(z)<0.) z=-z;
DB x=(Re(z)-2.)/12.; DB y=Im(z)/19.;
if(x*x+y*y<1.) return BesselJ0o(z);
               return BesselJ0B(z); }

// The function BesselJ0 returns of order of a dozen of correct decimal digits.

Drawing of curves

Files ado.cin, conto.cin and the besselj0.cin above should be loaded in the working directory for the compilation of the code below:

#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
using namespace std;
#include <complex>
typedef complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
#include "besselj0.cin"
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=801,M1=M+1;
int N=401,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("besselj0map1.eps","w");ado(o,162,82);
fprintf(o,"81 41 translate\n 10 10 scale\n"); 
DO(m,400) X[m]=-8.+.02*m;
     X[400]=-.001;
     X[401]= .001;
for(m=402;m<M1;m++) X[m]=-8.+.02*(m-1.);
DO(n,200)Y[n]=-4.+.02*n;
      Y[200]=-.001;
      Y[201]= .001;
for(n=202;n<N1;n++) Y[n]=-4.+.02*(n-1.);
//DO(m,M1)X[m]=Y[m];
for(m=-8;m<9;m++){if(m==0){M(m,-4.1)L(m,4.1)} else{M(m,-4)L(m,4)}}
for(n=-4;n<5;n++){     M(  -8,n)L(8,n)}
fprintf(o,".01 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=BesselJ0(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;
}
                     }}
//#include "plodi.cin"
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.5;q=.5;
for(m=-12;m<12;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".01 W 0 .6 0 RGB S\n");
for(m=0;m<12;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".01 W .9 0 0 RGB S\n");
for(m=0;m<12;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".01 W 0 0 .9 RGB S\n");
for(m=1;m<12;m++)  conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".02 W .9 0 0 RGB S\n");
for(m=1;m<12;m++)  conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 W 0 0 .9 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0.  ),-2*p,2*p); fprintf(o,".02 W .6 0 .6 RGB S\n");
for(m=-11;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
    m=0;          conto(o,g,w,v,X,Y,M,N, (0.+m),-2*p,2*p); fprintf(o,".02 W 0 0 0 RGB S\n");
for(m=1;m<12;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
//#include "plofu.cin"
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
     system("epstopdf besselj0map1.eps");
     system(    "open besselj0map1.pdf");            
     getchar(); system("killall Preview");//for mac
}

Licensing

   

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For any reuse or distribution, you must make clear to others the licence terms of this work (the best way to do this is with a link to this licence's web page). Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this licence impairs or restricts the author's moral rights.
Read the full licence.

References