imported>Dmitrii Kouznetsov |
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| == Summary == | | == Summary == |
| {{Image_Details|user
| | Importing file |
| |description = [[Complex map]] of the [[Bessel function]], <math>u+\mathrm i v=J_1(x+\mathrm i y)</math>
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| |author = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
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| |date-created = 2012 April 29
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| |pub-country = [[Japan]]
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| |notes = Copy from
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| <ref name="besselj0map">
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| http://tori.ils.uec.ac.jp/TORI/index.php/File:Besselj0map1T080.png
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| Complex map of the Bessel function BesselJ0
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| f=BesselJ0(x+iy) is shown in the x,y plane with lines u=ℜ(f)=const and lines v=ℑ(f)=const.</ref>
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| | |
| </ref>
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| |versions = http://tori.ils.uec.ac.jp/TORI/index.php/File:Besselj0map1T080.png
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| }}
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| | |
| ==Generators==
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| ==Implementation of [[BesselJ1]]==
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| // The file below should be stored as [[besselj1.cin]] in the working directory
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| | |
| // BesselJ[1,z] implementation:
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| z_type BesselJ1o(z_type z){ int n; z_type c,s,t;
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| s=1.; c=1.; t=-z*z/4.;
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| for(n=1;n<30;n++) {c/=0.+n*n; c*=t; s+=c/(1.+n);}
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| return s*z/2.;}
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| | |
| z_type BesselJ1B(z_type z){ int n; z_type a,f,c,s,t,u,v;
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| f=M_PI/4.+z; c=cos(f); s=sin(f); a=sqrt(2./M_PI/z); t=1./(z*z);
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| u= (( (( (( (( (
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| 11587123363874217382000061956546875./302231454903657293676544.*t
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| - 531595142508493798628972203125./1180591620717411303424.)*t
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| + 61394155884765868567528125./9223372036854775808.)*t
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| -1149690375852815671875./9007199254740992.)*t
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| + 232376754295310625./70368744177664.) * t
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| - 33424574007825./274877906944. ) * t
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| + 14783093325./2147483648. ) * t
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| - 2837835./4194304. ) * t
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| + 4725./32768. ) * t
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| - 15./128. ) *t
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| -1. ;
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| v= (((((((( (
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| - 38190914185478633427818266171875./9444732965739290427392.*t
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| +3918391713821821611515765625./73786976294838206464.)*t
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| -64152722972587114490625./72057594037927936.)*t
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| + 11100458801337530625./562949953421312.)*t
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| - 1327867167401775./2199023255552. )*t
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| + 468131288625./17179869184. )*t
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| - 66891825./33554432. )*t
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| + 72765./262144. )*t
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| - 105./1024. )*t
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| + 3./8. )/z;
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| return (c*u+s*v)*a;
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| }
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| | |
| z_type BesselJ1(z_type z){ DB x,y; x=Re(z)/12.6; y=Im(z)/18.5;
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| if(x*x+y*y<1.) return BesselJ1o(z);
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| if(x<0) return - BesselJ1B(-z);
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| return BesselJ1B(z);
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| }
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| // Copyleft 2012 by Dmitrii Kouznetsov.
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| // The code above is supposed to return $J_1(z)$ with a dozen correct decimal digits for any complex(double) $z$
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| | |
| ==Generator of curves==
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| 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.
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| | |
| #include <stdio.h>
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| #include <stdlib.h>
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| #define DB double
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| #define DO(x,y) for(x=0;x<y;x++)
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| using namespace std;
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| #include <complex>
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| typedef complex<double> z_type;
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| #define Re(x) x.real()
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| #define Im(x) x.imag()
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| #define I z_type(0.,1.)
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| #include "conto.cin"
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| #include "besselj1.cin"
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| | |
| main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
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| int M=801,M1=M+1;
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| int N=401,N1=N+1;
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| DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
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| char v[M1*N1]; // v is working array
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| FILE *o;o=fopen("besselj1map.eps","w");ado(o,162,82);
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| fprintf(o,"81 41 translate\n 10 10 scale\n");
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| DO(m,400) X[m]=-8.+.02*m;
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| X[400]=-.001;
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| X[401]= .001;
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| for(m=402;m<M1;m++) X[m]=-8.+.02*(m-1.);
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| DO(n,200)Y[n]=-4.+.02*n;
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| Y[200]=-.001;
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| Y[201]= .001;
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| for(n=202;n<N1;n++) Y[n]=-4.+.02*(n-1.);
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| //DO(m,M1)X[m]=Y[m];
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| for(m=-8;m<9;m++){if(m==0){M(m,-4.1)L(m,4.1)} else{M(m,-4)L(m,4)}}
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| for(n=-4;n<5;n++){ M( -8,n)L(8,n)}
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| fprintf(o,".01 W 0 0 0 RGB S\n");
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| DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
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| DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
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| DO(n,N1){y=Y[n]; z=z_type(x,y);
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| c=BesselJ1(z);
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| p=Re(c); q=Im(c);
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| if(p>-99. && p<99.
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| && q>-99. && q<99
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| )
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| {g[m*N1+n]=p;
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| f[m*N1+n]=q;
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| }
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| }}
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| //#include "plodi.cin"
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| fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.5;q=.5;
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| 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");
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| 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");
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| 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");
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| 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");
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| 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");
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| conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".02 W .6 0 .6 RGB S\n");
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| 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");
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| 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");
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| 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");
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| //#include "plofu.cin"
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| fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
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| system("epstopdf besselj1map.eps");
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| system( "open besselj1map.pdf");
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| getchar(); system("killall Preview");//for mac
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| }
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| // Copyleft 2012 by Dmitrii Kouznetsov
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| | |
| ==Generator of labels==
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| 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.
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| <nowiki>
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| \documentclass[12pt]{article} % <br>
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| \paperheight 838px % <br>
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| \paperwidth 1644px % <br>
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| \textwidth 1294px % <br>
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| \textheight 1200px % <br>
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| \topmargin -80px % <br>
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| \oddsidemargin -80px % <br>
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| \usepackage{graphics} % <br>
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| \usepackage{rotating} % <br>
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| \usepackage{color}% <br>
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| \newcommand \sx {\scalebox} % <br>
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| \newcommand \rot {\begin{rotate}} % <br>
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| \newcommand \ero {\end{rotate}} % <br>
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| \newcommand \ing {\includegraphics} % <br>
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| \newcommand \rmi {\mathrm{i}} % <br>
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| \begin{document} % <br>
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| \newcommand \zoomax { % <br>
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| \put(16,820){\sx{4.4}{$y$}} % <br>
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| \put(16,630){\sx{4}{$2$}} % <br>
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| \put(16,430){\sx{4}{$0$}} % <br>
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| \put(-4, 230){\sx{4}{$-\!2$}} % <br>
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| \put(220, 5){\sx{4}{$-\!6$}} % <br>
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| \put(420, 5){\sx{4}{$-\!4$}} % <br>
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| \put(620, 5){\sx{4}{$-\!2$}} % <br>
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| \put(843, 5){\sx{4}{$0$}} % <br>
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| \put(1043, 5){\sx{4}{$2$}} % <br>
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| \put(1243, 5){\sx{4}{$4$}} % <br>
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| \put(1443, 5){\sx{4}{$6$}} % <br>
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| \put(1631,6){\sx{4}{$x$}} % <br>
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| } % <br>
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| \parindent 0pt % <br>
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| \begin{picture}(1616,816) % <br>
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| %\put(40,30){\sx{10}{\ing{besselj1mapsmall}}} % small and fast version for debugging <br>
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| \put(40,30){\sx{10}{\ing{besselj1map}}} % <br>
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| %\put(40,30){\sx{10}{\ing{10}}} %draft <br>
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| \zoomax % <br>
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| \put(270,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} % <br>
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| \put(600,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} % <br>
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| \put(990,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} % <br>
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| \put(1340,430){\sx{4}{\rot{0}$v\!=\!0$\ero}} % <br>
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| \put(450,602){\sx{4}{\rot{0}$v\!=\!1$\ero}} \put(780,590){\sx{4}{\rot{0}$v\!=\!-1$\ero}} % <br>
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| \put(450,386){\sx{3}{\rot{0}$v\!=\!0.2$\ero}} \put(780,389){\sx{3}{\rot{0}$v\!=\!-0.2$\ero}} % <br>
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| \put(450,344){\sx{3}{\rot{0}$v\!=\!0.4$\ero}} \put(780,356){\sx{3}{\rot{0}$v\!=\!-0.4$\ero}} % <br>
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| \put(161,480){\sx{4}{\rot{86}$u\!=\!0$\ero}} % <br>
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| \put(480,480){\sx{4}{\rot{80}$u\!=\!0$\ero}} % <br>
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| \put(775,460){\sx{3}{\rot{72}$u\!=\!0.4$\ero}} % <br>
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| \put(865,472){\sx{4}{\rot{90}$u\!=\!0$\ero}} % <br>
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| \put(949,455){\sx{3}{\rot{109}$u\!=\!-0.4$\ero}} % <br>
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| \put(1250,480){\sx{4}{\rot{99}$u\!=\!0$\ero}} % <br>
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| \put(1569,480){\sx{4}{\rot{95}$u\!=\!0$\ero}} % <br>
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| \end{picture} <br>
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| \end{document} % <br>
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| | |
| %Copyleft 2012 by Dmitrii Kouznetsov
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| </nowiki>
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| | |
| ==Generator of png==
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| Unfortunately, CZ does not allow to use the PDF images.
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| The [[besselj1mapT.pdf]] generated withe the code above is converted to the PNG format with resolution 80 pixels/inch.
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| == Licensing ==
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| {{CC|by|3.0}}
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| ==References==
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| <references/>
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