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_0(x+\mathrm i y)</math>j_
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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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| ===besselj0.cin===
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| The complex(double) [[C++]] implementation [[besselj0.cin]] is used:
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| // The code below should be stored in the working directory as [[besselj0.cin]]
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| z_type BesselJ0o(z_type z){ int n; z_type c,s,t;
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| s=1.; c=1.; t=-z*z/4.; for(n=1;n<32;n++) {c/=0.+n*n; c*=t; s+=c;}
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| return s;}
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| z_type BesselJ0B(z_type z){ int n; z_type c,C,s,S,t,u,x;
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| t=M_PI/4.-z; c=cos(t); s=sin(t); u=1./16./(z*z);
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| C=(((((((((((
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| + 11021897833929133607268351617203125./137438953472.)*u
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| - 502860269940467106811189921875./8589934592.)*u
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| + 57673297952355815927071875./1073741824.)*u
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| - 1070401384414690453125./16777216.)*u
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| + 213786613951685775./2097152.)*u
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| - 30241281245175./131072.)*u
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| + 13043905875./16384.)*u
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| - 2401245./512.)*u
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| + 3675./64.)* u
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| - 9./4.)*u + 2.)* c;
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| S=(((((((((((
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| - 882276678992136837800861860405640625./274877906944.)*u
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| + 36232405765710498380237842265625./17179869184.)*u
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| - 3694483615889146090857721875./2147483648.)*u
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| + 60013837619516978071875./33554432.)*u
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| - 10278202593831046875./4194304.)*u
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| + 1212400457192925./262144.)*u
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| - 418854310875./32768.)*u
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| + 57972915./1024.)*u
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| - 59535./128.)*u
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| + 75./8.)*u - 1.) *s/4./z;
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| return (C+S)/sqrt(2.*M_PI*z);}
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| z_type BesselJ0(z_type z){ if(Re(z)<0.) z=-z;
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| DB x=(Re(z)-2.)/12.; DB y=Im(z)/19.;
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| if(x*x+y*y<1.) return BesselJ0o(z);
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| return BesselJ0B(z); }
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| // The function BesselJ0 returns of order of a dozen of correct decimal digits.
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| ===Drawing of curves===
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| Files [[ado.cin]], [[conto.cin]] and the [[besselj0.cin]] above should be loaded in the working directory for the compilation of the code below:
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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 "besselj0.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("besselj0map1.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=BesselJ0(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 besselj0map1.eps");
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| system( "open besselj0map1.pdf");
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| getchar(); system("killall Preview");//for mac
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| }
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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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