File:Besselj1mapT080.png

Summary

Title / Description	Complex map of the Bessel function, ${\displaystyle u+\mathrm {i} v=J_{1}(x+\mathrm {i} y)}$
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]
Other versions	http://tori.ils.uec.ac.jp/TORI/index.php/File:Besselj0map1T080.png
Using this image on CZ	Please click here to add the credit line, then copy the code below to add this image to a Citizendium article, changing the size, alignment, and caption as necessary. {{Image|Besselj1mapT080.png|right|350px|Add image caption here.}}

Generators

Implementation of BesselJ1

// The code below should be stored as besselj1.cin in the working directory

// BesselJ[1,z] implementation:
z_type BesselJ1o(z_type z){ int n; z_type c,s,t;
s=1.; c=1.; t=-z*z/4.; 
for(n=1;n<30;n++) {c/=0.+n*n; c*=t; s+=c/(1.+n);}
return s*z/2.;}

z_type BesselJ1B(z_type z){ int n; z_type a,f,c,s,t,u,v;
f=M_PI/4.+z; c=cos(f); s=sin(f); a=sqrt(2./M_PI/z); t=1./(z*z);
u= (( (( (( (( (
11587123363874217382000061956546875./302231454903657293676544.*t
- 531595142508493798628972203125./1180591620717411303424.)*t
  + 61394155884765868567528125./9223372036854775808.)*t
     -1149690375852815671875./9007199254740992.)*t
       + 232376754295310625./70368744177664.) * t
          - 33424574007825./274877906944. ) * t
            + 14783093325./2147483648.   ) * t
               - 2837835./4194304.      ) * t
                 + 4725./32768.        ) * t
                  - 15./128.          ) *t
                    -1. ;
v= (((((((( (
- 38190914185478633427818266171875./9444732965739290427392.*t
+3918391713821821611515765625./73786976294838206464.)*t
  -64152722972587114490625./72057594037927936.)*t
   + 11100458801337530625./562949953421312.)*t
      - 1327867167401775./2199023255552. )*t
         + 468131288625./17179869184.   )*t
            - 66891825./33554432.      )*t
              + 72765./262144.        )*t
               - 105./1024.          )*t
              + 3./8.             )/z;
return (c*u+s*v)*a;
}

z_type BesselJ1(z_type z){ DB x,y; x=Re(z)/12.6; y=Im(z)/18.5;
if(x*x+y*y<1.) return BesselJ1o(z);
if(x<0) return - BesselJ1B(-z);
        return   BesselJ1B(z);
}
// Copyleft 2012 by Dmitrii Kouznetsov. 
// The code above is supposed to return $J_1(z)$ with a dozen correct decimal digits for any complex(double) $z$

Generator of curves

File besselj1.cin above and files ado.cin and conto.cin should be loaded to the working directory in order to compile the C++ code below. The name besselj1map.cc is suggested for this code.

#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 "besselj1.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("besselj1map.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=BesselJ1(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 besselj1map.eps");
     system(    "open besselj1map.pdf");            
     getchar(); system("killall Preview");//for mac
}

// Copyleft 2012 by Dmitrii Kouznetsov

Generator of labels

File besselj1map.pdf should be generated with the C++ code above and stored in the working directory; then the Latex document below (let it be stored as Besselj1mapT.tex ) can be compiled. \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{besselj1map}}} % <br> \zoomax % <br> \put(270,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} % <br> \put(600,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} % <br> \put(990,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} % <br> \put(1340,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} % <br> \put(796,656){\sx{4}{\rot{0}$v\!=\!2$\ero}} % <br> \put(440,602){\sx{4}{\rot{0}$v\!=\!-1$\ero}} \put(796,586){\sx{4}{\rot{0}$v\!=\!1$\ero}} % <br> \put(450,262){\sx{4}{\rot{0}$v\!=\!1$\ero}}  %<br> \put(450,190){\sx{4}{\rot{0}$v\!=\!2$\ero}} % <br> \put(450,386){\sx{3}{\rot{0}$v\!=\!0.2$\ero}} \put(780,389){\sx{3}{\rot{0}$v\!=\!-0.2$\ero}} % <br> \put(450,344){\sx{3}{\rot{0}$v\!=\!0.4$\ero}} \put(780,356){\sx{3}{\rot{0}$v\!=\!-0.4$\ero}} % <br> \put(161,480){\sx{4}{\rot{86}$u\!=\!0$\ero}} % <br> \put(480,480){\sx{4}{\rot{80}$u\!=\!0$\ero}} % <br> \put(770,452){\sx{3}{\rot{72}$u\!=\!-0.4$\ero}} % <br> \put(1064,534){\sx{3}{\rot{36}$u\!=\!0.8$\ero}} % <br> \put(1080,482){\sx{3}{\rot{48}$u\!=\!0.6$\ero}} % <br> \put(1142,452){\sx{3}{\rot{72}$u\!=\!0.4$\ero}} % <br> \put(865,472){\sx{4}{\rot{90}$u\!=\!0$\ero}} % <br> \put(949,468){\sx{3}{\rot{109}$u\!=\!0.4$\ero}} % <br> \put(1250,480){\sx{4}{\rot{99}$u\!=\!0$\ero}} % <br> \put(1569,480){\sx{4}{\rot{95}$u\!=\!0$\ero}} % <br> \end{picture}  %<br> \end{document} % <br> %Copyleft 2012 by Dmitrii Kouznetsov

Generator of png

Unfortunately, CZ does not allow to use the PDF images. The besselj1mapT.pdf generated withe the code above is converted to the PNG format with resolution 80 pixels/inch.

Licensing

   

This media, Besselj1mapT080.png, is licenced under the Creative Commons Attribution 3.0 Unported License

You are free: To Share — To copy, distribute and transmit the work; To Remix — To adapt the work.
Under the following conditions: Attribution — You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).
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