File:BesselH0mapT050.png

Complex mp of the Hankel function of zeroth order, id est, map of BesselH0 in the complex plane.

$ u+\mathrm i v = \mathrm{BesselH}[0,x+\mathrm i y]$

C++ generator of curves

// Files ado.cin, conto.cin, besselj0.cin, bessely0.cin should be loaded at the working directory in order to compile the C++ 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"
#include "bessely0.cin"

main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=803,M1=M+1;
int N=201,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("besselH0map.eps","w");ado(o,202,102);
fprintf(o,"101 51 translate\n 10 10 scale\n"); 

/* fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.5;q=.5;

fprintf(o,"0  2.5 13 -11 11 arc  .3 W .6 1 1 RGB S\n");
fprintf(o,"0  2.5 13 159 191 arc   S\n");
fprintf(o,"0 -2.5 13 -11 11 arc  S\n");
fprintf(o,"0 -2.5 13 -191 -159 arc  S\n");

• /

DO(m,400) X[m]=-10.+.025*m;
    X[400]=-.014;
    X[401]=-.004;
    X[402]= .004;
    X[403]= .014;
for(m=404;m<M1;m++) X[m]=-10.+.025*(m-3.);
DO(n,100)Y[n]=-5.+.05*n;
     Y[100]=-.006;
     Y[101]= .006;
for(n=102;n<N1;n++) Y[n]=-5.+.05*(n-1.);
//DO(m,M1)X[m]=Y[m];
for(m=-10;m<11;m++){if(m==0){M(m,-5.1)L(m,5.1)} else{M(m,-5)L(m,5)}}
for(n=-5;n<6;n++){     M(  -10,n)L(10,n)}
fprintf(o,"0 setlinecap .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);
 //   d=BesselY0o(z);
    c=BesselJ0(z) + I * BesselY0(z);
// p= -log(abs(c-d)/abs(c+d))/log(10.);
   p=Re(c);    q=Im(c);        
    if(p>-30. && p<30.
     &&     q>-30. && q<30. 
    )
{g[m*N1+n]=p; 
 f[m*N1+n]=q;
}
                     }}
//#include "plodi.cin"
/*
conto(o,g,w,v,X,Y,M,N,15, -p,p); fprintf(o,".07 W 1 0 1 RGB S\n");
conto(o,g,w,v,X,Y,M,N,14, -p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,13, -p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,12, -p,p); fprintf(o,".06 W 0 0 1 RGB S\n");
conto(o,g,w,v,X,Y,M,N,11, -p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,10, -p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 9, -p,p); fprintf(o,".06 W 0 1 1 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 8, -p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 7, -p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 6, -p,p); fprintf(o,".07 W 0 1 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 5, -p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 4, -p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 3, -p,p); fprintf(o,".06 W .5 .5 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 2, -p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 1, -p,p); fprintf(o,".1 W 1 0 0 RGB S\n");
*/
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=2;q=.8;
for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".016 W 0 .6 0 RGB S\n");
for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".016 W .9 0 0 RGB S\n");
for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".016 W 0 0 .9 RGB S\n");
for(m=1;m<21;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".03 W .9 0 0 RGB S\n");
for(m=1;m<21;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 W 0 0 .9 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0.  ),-2*p,2*p); fprintf(o,".03 W .6 0 .6 RGB S\n");
for(m=-20;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 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,".03 W 0 0 0 RGB S\n");
for(m=1;m<21;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 W 0 0 0 RGB S\n");
//#include "plofu.cin"
M(0,0)L(-20,0) fprintf(o,"1 1 1 RGB 0.032 W 0 setlinecap S\n"); 
DO(n,80) {x=-.25*n; M(x-.10,0) L(x-.20,0)} fprintf(o,"0 0 0 RGB .04 W S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
    system("epstopdf besselH0map.eps");
    system(    "open besselH0map.pdf");            
    getchar(); system("killall Preview");//for mac
}

Latex generator of labels

% File besselH0map.pdf should be generated with the code above in order to compile the Latex document below.

% The document should be saved as BesselH0mapT.tex
% The compilation gives BesselH0mapT.pdf
% that file can be converted to PNG format with resolution 50  picels/inch giving the file loaded.
% Copyleft 2012 by Dmitrii Kouznetsov.

% \documentclass[12pt]{article} %<br> \paperheight 1038px %<br> \paperwidth 2044px %<br> \textwidth 2094px %<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(16,1020){\sx{4.4}{$y$}} %<br> \put(16,930){\sx{4}{$4$}} %<br> \put(16,830){\sx{4}{$3$}} %<br> \put(16,730){\sx{4}{$2$}} %<br> \put(16,630){\sx{4}{$1$}} %<br> \put(16,530){\sx{4}{$0$}} %<br> \put(-4, 430){\sx{4}{$-\!1$}} %<br> \put(-4, 330){\sx{4}{$-\!2$}} %<br> \put(-4, 230){\sx{4}{$-\!3$}} %<br> \put(-4, 130){\sx{4}{$-\!4$}} %<br> %<br> \put(220, 5){\sx{4}{$-\!8$}} %<br> \put(420, 5){\sx{4}{$-\!6$}} %<br> \put(620, 5){\sx{4}{$-\!4$}} %<br> \put(820, 5){\sx{4}{$-\!2$}} %<br> \put(1043, 5){\sx{4}{$0$}} %<br> \put(1243, 5){\sx{4}{$2$}} %<br> \put(1443, 5){\sx{4}{$4$}} %<br> \put(1643, 5){\sx{4}{$6$}} %<br> \put(1843, 5){\sx{4}{$8$}} %<br> \put(2031,6){\sx{4}{$x$}} %<br> %<br> } %<br> \parindent 0pt %<br> \begin{picture}(2016,1016) %<br> %\put(40,30){\sx{10}{\ing{besselj1o}}} %<br> %\put(40,30){\sx{10}{\ing{besselY0mapWide}}} %<br> %\put(40,30){\sx{10}{\ing{besselH0small}}} %<br> \put(40,30){\sx{10}{\ing{besselH0map}}} %<br> \zoomax %<br> %<br> \put( 100,526){\sx{6}{\bf cut}} %<br> %<br> \put( 1070,880){\sx{5}{\rot{90}$u\!=\!0$\ero}} %<br> \put( 1208,880){\sx{5}{\rot{90}$v\!=\!0$\ero}} %<br> \multiput( 182,880)(300,0){3}{\sx{5}{\rot{90}$u\!=\!0$\ero}} %<br> \multiput( 335,880)(299,0){3}{\sx{5}{\rot{90}$v\!=\!0$\ero}} %<br> \multiput(1354,880)(302,0){3}{\sx{5}{\rot{90}$u\!=\!0$\ero}} %<br> \multiput(1504,880)(300,0){3}{\sx{5}{\rot{90}$v\!=\!0$\ero}} %<br> \put( 264,570){\sx{3.6}{$u\!=\!-0.2$}} %<br> \put( 426,580){\sx{3.6}{$v\!=\!-0.2$}} %<br> \put( 590,600){\sx{3.6}{$u\!=\!0.2$}} %<br> \put( 736,614){\sx{3.6}{$v\!=\!0.2$}} %<br> \put( 860,652){\sx{3.6}{$u\!=\!-0.2$}} %<br> \put( 1106,646){\sx{3.6}{$u\!=\!0.2$}} %<br> \put( 1234,622){\sx{3.6}{$v\!=\!0.2$}} %<br> \put( 1370,606){\sx{3.6}{$u\!=\!-0.2$}} %<br> \put( 1542,588){\sx{3.6}{$v\!=\!-0.2$}} %<br> \put( 1724,570){\sx{3.6}{$u\!=\!0.2$}} %<br> \put( 1880,564){\sx{3.6}{$v\!=\!0.2$}} %<br> \put( 202,494){\sx{5}{$v\!=\!0$}} %<br> \put( 512,496){\sx{5}{$v\!=\!0$}} %<br> \put( 820,498){\sx{5}{$v\!=\!0$}} %<br> \put( 455,402){\sx{5}{$v\!=\!1$}} %<br> \put( 780,434){\sx{5}{$v\!=\!-1$}} %<br> \put( 780,376){\sx{5}{$v\!=\!-2$}} %<br> \put( 990,390){\sx{5}{$u\!=\!3$}} %<br> \put( 990,340){\sx{5}{$u\!=\!4$}} %<br> \put(1198,464){\sx{5}{$v\!=\!1$}} %<br> \put(1190,390){\sx{5}{$v\!=\!2$}} %<br> \end{picture} % %<br> \end{document} % %<br> % % Copyleft 2012 by Dmitrii Kouznetsov

Keywords

Besselj0.cin, Bessely0.cin, ado.cin, conto.cin, BesselH0, C++, Latex, Complex map

References

http://people.math.sfu.ca/~cbm/aands/page_359.htm
http://en.wikipedia.org/wiki/Bessel_function
http://mathworld.wolfram.com/HankelFunction.html
http://mathworld.wolfram.com/HankelFunctionoftheFirstKind.html
http://www.mathworks.co.jp/help/techdoc/ref/besselh.html
http://dlmf.nist.gov/10.3