File:Besselj1mapT080.png

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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References