File:Besselj0map1T080.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 $v\!=\!\Im(f)\!=\!\mathrm{const}$.

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.

using namespace std; typedef complex z_type; 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-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 }
 * 1) include 
 * 2) include 
 * 3) define DB double
 * 4) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "conto.cin"
 * 5) include "besselj0.cin"

Latex generator of labels
% File acoscmap.pdf should be generated with the code above in order to compile the latex document below: %

\documentclass[12pt]{article} % \paperheight 838px % \paperwidth 1644px % \textwidth 1294px % \textheight 1200px % \topmargin -80px % \oddsidemargin -80px % \usepackage{graphics} % \usepackage{rotating} % \usepackage{color}% \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \newcommand \rmi {\mathrm{i}} % \begin{document} % \newcommand \zoomax { % \put(16,820){\sx{4.4}{$y$}} % \put(16,630){\sx{4}{$2$}} % \put(16,430){\sx{4}{$0$}} % \put(-4, 230){\sx{4}{$-\!2$}} % \put(220, 5){\sx{4}{$-\!6$}} % \put(420, 5){\sx{4}{$-\!4$}} % \put(620, 5){\sx{4}{$-\!2$}} % \put(843, 5){\sx{4}{$0$}} % \put(1043, 5){\sx{4}{$2$}} % \put(1243, 5){\sx{4}{$4$}} % \put(1443, 5){\sx{4}{$6$}} % \put(1631,6){\sx{4}{$x$}} % } % \parindent 0pt % \begin{picture}(1616,816) % \put(40,30){\sx{10}{\ing{besselj0map1}}} % \zoomax % \put(170,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} % \put(800,613){\sx{4}{$u\!=\!2$}} % \put(865,500){\sx{4}{\rot{90}$v\!=\!0$\ero}} % \put(872,446){\sx{4}{\rot{46}$u\!=\!1$\ero}} % \put(862,420){\sx{4}{\rot{-46}$u\!=\!1$\ero}} % \put(630,468){\sx{4}{\rot{0}$v\!=\!0.2$\ero}} % \put(630,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} \put(900,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} % \put(628,392){\sx{4}{\rot{0}$v\!=\!-0.2$\ero}} % \put(624,390){\sx{4}{\rot{90}$u\!=\!0$\ero}} % \put(160,390){\sx{4}{\rot{90}$v\!=\!0$\ero}} % \put(312,396){\sx{4}{\rot{90}$u\!=\!0$\ero}} % \put(362,366){\sx{3.8}{\rot{90}$u\!=\!-0.2$\ero}} % \put(428,370){\sx{3.8}{\rot{90}$u\!=\!-0.4$\ero}} % \put(482,396){\sx{4}{\rot{90}$v\!=\!0$\ero}} % \put(1106,390){\sx{4}{\rot{90}$u\!=\!0$\ero}} % \put(1250,390){\sx{4}{\rot{90}$v\!=\!0$\ero}} % \put(1419,390){\sx{4}{\rot{90}$u\!=\!0$\ero}} % \put(1568,390){\sx{4}{\rot{90}$v\!=\!0$\ero}} % \end{picture} % \end{document} % %