Difference between revisions of "File:Besselj0map1T080.png"

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[[Complex map]] of the Bessel function [[BesselJ0]]
−
Importing image file
  
  +
  +
$f=\mathrm{BesselJ}_0(x+\mathrm i y)$ is shown in the $x$,$y$ plane with
  +
lines $u\!=\!\Re(f)\!=\!\mathrm{const}$ and
  +
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 [[ado.cin]] and [[conto.cin]] should be stored in the working directory in order to compile 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
  +
}
  +
  +
==Latex generator of labels==
  +
  +
% File [[acoscmap.pdf]] should be generated with the code above in order to compile the [[latex]] document below:
  +
%<nowiki><br>
  +
  +
\documentclass[12pt]{article} %<br>
  +
\paperheight 838px %<br>
  +
\paperwidth 1644px %<br>
  +
\textwidth 1294px %<br>
  +
\textheight 1200px %<br>
  +
\topmargin -80px %<br>
  +
\oddsidemargin -80px %<br>
  +
\usepackage{graphics} %<br>
  +
\usepackage{rotating} %<br>
  +
\usepackage{color}%<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}{$-\!6$}} %<br>
  +
\put(420, 5){\sx{4}{$-\!4$}} %<br>
  +
\put(620, 5){\sx{4}{$-\!2$}} %<br>
  +
\put(843, 5){\sx{4}{$0$}} %<br>
  +
\put(1043, 5){\sx{4}{$2$}} %<br>
  +
\put(1243, 5){\sx{4}{$4$}} %<br>
  +
\put(1443, 5){\sx{4}{$6$}} %<br>
  +
\put(1631,6){\sx{4}{$x$}} %<br>
  +
} %<br>
  +
\parindent 0pt %<br>
  +
\begin{picture}(1616,816) %<br>
  +
\put(40,30){\sx{10}{\ing{besselj0map1}}} %<br>
  +
\zoomax %<br>
  +
\put(170,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} %<br>
  +
\put(800,613){\sx{4}{$u\!=\!2$}} %<br>
  +
\put(865,500){\sx{4}{\rot{90}$v\!=\!0$\ero}} %<br>
  +
\put(872,446){\sx{4}{\rot{46}$u\!=\!1$\ero}} %<br>
  +
\put(862,420){\sx{4}{\rot{-46}$u\!=\!1$\ero}} %<br>
  +
\put(630,468){\sx{4}{\rot{0}$v\!=\!0.2$\ero}} %<br>
  +
\put(630,430){\sx{4}{\rot{0}$v\!=\!0$\ero}}
  +
\put(900,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} %<br>
  +
\put(628,392){\sx{4}{\rot{0}$v\!=\!-0.2$\ero}} %<br>
  +
\put(624,390){\sx{4}{\rot{90}$u\!=\!0$\ero}} %<br>
  +
\put(160,390){\sx{4}{\rot{90}$v\!=\!0$\ero}} %<br>
  +
\put(312,396){\sx{4}{\rot{90}$u\!=\!0$\ero}} %<br>
  +
\put(362,366){\sx{3.8}{\rot{90}$u\!=\!-0.2$\ero}} %<br>
  +
\put(428,370){\sx{3.8}{\rot{90}$u\!=\!-0.4$\ero}} %<br>
  +
\put(482,396){\sx{4}{\rot{90}$v\!=\!0$\ero}} %<br>
  +
\put(1106,390){\sx{4}{\rot{90}$u\!=\!0$\ero}} %<br>
  +
\put(1250,390){\sx{4}{\rot{90}$v\!=\!0$\ero}} %<br>
  +
\put(1419,390){\sx{4}{\rot{90}$u\!=\!0$\ero}} %<br>
  +
\put(1568,390){\sx{4}{\rot{90}$v\!=\!0$\ero}} %<br>
  +
\end{picture} %<br>
  +
\end{document} %<br>
  +
%</nowiki>
  +
  +
==References==
  +
  +
http://reference.wolfram.com/mathematica/ref/BesselJ.ja.html
  +
  +
[[Category:BesselJ0]]
  +
[[Category:Complex map]]
  +
[[Category:Bessel function]]
  +
[[Category:Cylindric function]]
  +
[[Category:C++]]
  +
[[Category:Latex]]

Latest revision as of 09:41, 21 June 2013

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 $v\!=\!\Im(f)\!=\!\mathrm{const}$.

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 ado.cin and conto.cin should be stored in the working directory in order to compile 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
}

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 1644px %<br> \textwidth 1294px %<br> \textheight 1200px %<br> \topmargin -80px %<br> \oddsidemargin -80px %<br> \usepackage{graphics} %<br> \usepackage{rotating} %<br> \usepackage{color}%<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}{$-\!6$}} %<br> \put(420, 5){\sx{4}{$-\!4$}} %<br> \put(620, 5){\sx{4}{$-\!2$}} %<br> \put(843, 5){\sx{4}{$0$}} %<br> \put(1043, 5){\sx{4}{$2$}} %<br> \put(1243, 5){\sx{4}{$4$}} %<br> \put(1443, 5){\sx{4}{$6$}} %<br> \put(1631,6){\sx{4}{$x$}} %<br> } %<br> \parindent 0pt %<br> \begin{picture}(1616,816) %<br> \put(40,30){\sx{10}{\ing{besselj0map1}}} %<br> \zoomax %<br> \put(170,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} %<br> \put(800,613){\sx{4}{$u\!=\!2$}} %<br> \put(865,500){\sx{4}{\rot{90}$v\!=\!0$\ero}} %<br> \put(872,446){\sx{4}{\rot{46}$u\!=\!1$\ero}} %<br> \put(862,420){\sx{4}{\rot{-46}$u\!=\!1$\ero}} %<br> \put(630,468){\sx{4}{\rot{0}$v\!=\!0.2$\ero}} %<br> \put(630,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} \put(900,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} %<br> \put(628,392){\sx{4}{\rot{0}$v\!=\!-0.2$\ero}} %<br> \put(624,390){\sx{4}{\rot{90}$u\!=\!0$\ero}} %<br> \put(160,390){\sx{4}{\rot{90}$v\!=\!0$\ero}} %<br> \put(312,396){\sx{4}{\rot{90}$u\!=\!0$\ero}} %<br> \put(362,366){\sx{3.8}{\rot{90}$u\!=\!-0.2$\ero}} %<br> \put(428,370){\sx{3.8}{\rot{90}$u\!=\!-0.4$\ero}} %<br> \put(482,396){\sx{4}{\rot{90}$v\!=\!0$\ero}} %<br> \put(1106,390){\sx{4}{\rot{90}$u\!=\!0$\ero}} %<br> \put(1250,390){\sx{4}{\rot{90}$v\!=\!0$\ero}} %<br> \put(1419,390){\sx{4}{\rot{90}$u\!=\!0$\ero}} %<br> \put(1568,390){\sx{4}{\rot{90}$v\!=\!0$\ero}} %<br> \end{picture} %<br> \end{document} %<br> %

References

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

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