File:Besselj0mapT100.png

Complex map of the Bessel function BesselJ0

$f=\mathrm{BesselJ}_0(x+\mathrm i y)$ is shown in the $x$,$y$ plane with lines $u\!=\!\Re(f)\!=\!\mathrm{const}$ and lines lines $v\!=\!\Im(f)\!=\!\mathrm{const}$.

C++ implementation of BesselJ0

// The file 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.

C++ generator of curves

// Fies besselj0.cin above and conto.cin should be stored in the working directory in order to compile the code below.

#include <math.h>
#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"

// 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<12;n++) {c/=0.+n*n; c*=t; s+=c;} return s;}

main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=401,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("besselj0map.eps","w");ado(o,82,82);
fprintf(o,"41 41 translate\n 10 10 scale\n");
DO(n,200)Y[n]=-4.+.02*n;
      Y[200]=-.002;
      Y[201]= .002;
for(n=202;n<N1;n++) Y[n]=-4.+.02*(n-1.);
DO(m,M1)X[m]=Y[m];
for(m=-4;m<5;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(  -4,n)L(4,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);
//        d= cosc(c);
//     p=abs((z-d)/(z+d));  p=-log(p)/log(10.);
       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;
       }
                      }}
// Draw the contours:
// #include "plodi.cin"
 #include "plofu.cin"
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
      system("epstopdf besselj0map.eps");
      system(    "open besselj0map.pdf");            
      getchar(); system("killall Preview");//for mac
}

Latex generator of labels

% File acoscmap.pdf should be generated with the code above in order to compile the latex document below: %<br> \documentclass[12pt]{article} %<br> \paperheight 838px %<br> \paperwidth 844px %<br> \textwidth 1294px %<br> \textheight 1200px %<br> \topmargin -80px %<br> \oddsidemargin -80px %<br> \usepackage{graphics} %<br> \usepackage{rotating} %<br> \newcommand \sx {\scalebox} %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \newcommand \ing {\includegraphics} %<br> \newcommand \rmi {\mathrm{i}} %<br> \begin{document} %<br> \newcommand \zoomax { %<br> \put(16,820){\sx{4.4}{$y$}} %<br> \put(16,630){\sx{4}{$2$}} %<br> \put(16,430){\sx{4}{$0$}} %<br> \put(-4, 230){\sx{4}{$-\!2$}} %<br> \put(220, 5){\sx{4}{$-\!2$}} %<br> \put(443, 5){\sx{4}{$0$}} %<br> \put(643, 5){\sx{4}{$2$}} %<br> \put(831,6){\sx{4}{$x$}} %<br> } %<br> \parindent 0pt %<br> %\sx{8}{\begin{picture}(86,86) \put(0,0){\ing{b271t0}} %<br> \begin{picture}(816,816) %<br> \put(40,30){\sx{10}{\ing{besselj0map}}} %<br> \zoomax %<br> \put(463,490){\sx{4}{\rot{90}$v\!=\!0$\ero}} %<br> \put(463,320){\sx{4}{\rot{90}$v\!=\!0$\ero}} %<br> \put(80,484){\sx{4}{\rot{86}$v\!=\!0$\ero}} %<br> \put(222,469){\sx{4}{\rot{77}$u\!=\!0$\ero}} %<br> \put(498,470){\sx{4}{\rot{49}$u\!=\!1$\ero}} %<br> \put(470,411){\sx{4}{\rot{-50}$u\!=\!1$\ero}} %<br> \put(480,430){\sx{4}{$v\!=\!0$}} %<br> \put(231,463){\sx{3.8}{\rot{1}$v\!=\!0.2$\ero}} %<br> \put(230,431){\sx{3.8}{$v\!=\!0$}} %<br> \put(226,399){\sx{3.8}{\rot{-5}$v\!=\!-0.2$\ero}} %<br> \end{picture} %<br> \end{document} %<br> %

References

http://reference.wolfram.com/mathematica/ref/BesselJ.ja.html