imported>Meg Taylor |
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| == Summary == | | == Summary == |
| {{Image_Details|user
| | Importing file |
| |description = [[Complex map]] of the Principal branch of the [[Lambert W function]]
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| |author = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
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| |date-created = 2011 September 25
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| |pub-country = Japan
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| |notes = The [[Tania function]] is used for the evaluation; files [[conto.cin]] and [[ado.cin]] are used for the plotting.
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| |versions = http://tori.ils.uec.ac.jp/TORI/index.php/File:LambertWmap150.png
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| }}
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| == Licensing ==
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| {{CC|by|3.0}}
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| ==Generators ==
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| First, the curves are generated with the [[C++]] program.
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| The principal branch of the Lambert W function is easy to evaluate through the [[Tania function]], the code is supplied below.
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| Then, the labels are added in the [[Latex]] document.
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| The result is converted to [[Portable Network Graphics|PNG]] or any other appropriate format.
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| ===[[C++]] generator of the curves===
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| Files [[conto.cin]] and [[ado.cin]] should be in the working directory for the compilation of the code below:
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| #include <math.h>
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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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| z_type ArcTania(z_type z) {return z + log(z) - 1. ;}
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| z_type ArcTaniap(z_type z) {return 1. + 1./z ;}
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| z_type TaniaTay(z_type z) { int n; z_type s;
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| s=1.+z*(.5+z*(1./16.+z*(-1./192.+z*(-1./3072.+z*(1.3/6144.+z*(-4.7/147456.
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| //+z*(7.3/4128768.) //some reserve term
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| )))))); DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
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| z_type TaniaNega(z_type z){int n;z_type s=exp(z-exp(z)+1.);
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| DO(n,4) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
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| z_type TaniaBig(z_type z){int n;z_type s=z; s=z-log(s)+1.;
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| DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
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| z_type TaniaS(z_type z){int n; z_type s,t=z+z_type(2.,-M_PI);t*=2./9.; t=I*sqrt(t);
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| s=-1.+t*(3.+t*(-3.+t*(.75+t*(.3+t*(.9/16.+t*(-.3/7.+t*(-12.51/224. //+t*(-.9/28.)
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| )))))));
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| DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
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| z_type Tania(z_type z){ z_type t;
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| if( fabs(Im(z))< M_PI && Re(z)<-2.51) return TaniaNega(z);
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| if( abs(z)>7. || Re(z)>3.8 ) return TaniaBig(z);
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| if( Im(z) > .7 ) return TaniaS(z);
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| if( Im(z) < -.7) return conj(TaniaS(conj(z)));
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| return TaniaTay(z);
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| }
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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=160,M1=M+1;
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| int N=161,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("productlogmap.eps","w");ado(o,162,162);
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| FILE *o;o=fopen("lambertwmap.eps","w");ado(o,162,162);
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| fprintf(o,"81 81 translate\n 10 10 scale\n");
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| DO(m,M1) X[m]=-8.+.1*(m);
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| DO(n,80)Y[n]=-8.+.1*n;
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| Y[80]=-.03;
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| Y[81]= .03;
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| for(n=82;n<N1;n++) Y[n]=-8.+.1*(n-1.);
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| for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}}
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| for(n=-8;n<9;n++){ M( -8,n)L(8,n)}
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| fprintf(o,".008 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=Tania(log(z)-1.); p=Re(c);q=Im(c);
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| if(p>-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
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| }}
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| fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=.6;q=.5;
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| 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,".03 W 0 .6 0 RGB S\n");
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| 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,".03 W .9 0 0 RGB S\n");
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| 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,".03 W 0 0 .9 RGB S\n");
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| for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".07 W .9 0 0 RGB S\n");
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| for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".07 W 0 0 .9 RGB S\n");
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| conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".07 W .6 0 .6 RGB S\n");
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| for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".07 W 0 0 0 RGB S\n");
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| y=0.; for(m=0;m<75;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)}
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| fprintf(o,".07 W 1 .6 0 RGB S\n");
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| y=0.; for(m=2;m<75;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)}
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| fprintf(o,".07 W 0 .6 1 RGB S\n");
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| fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
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| system("epstopdf lambertwmap.eps");
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| system( "open lambertwmap.pdf");
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| getchar(); system("killall Preview");
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| }
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| ===[[Latex]] document generating the labels===
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| \documentclass[12pt]{article} %
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| \usepackage{geometry} %
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| \usepackage{graphicx} %
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| \usepackage{rotating} %
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| \paperwidth 854pt %
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| \paperheight 844pt %
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| \topmargin -96pt %
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| \oddsidemargin -98pt %
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| \textwidth 1100pt %
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| \textheight 1100pt %
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| \pagestyle {empty} %
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| \newcommand \sx {\scalebox} %
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| \newcommand \rot {\begin{rotate}} %
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| \newcommand \ero {\end{rotate}} %
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| \newcommand \ing {\includegraphics} %
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| \begin{document} %
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| \sx{5}{ \begin{picture}(164,165) %
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| %\put(6,5){\ing{taniacontour}} %
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| %\put(6,5){\ing{productlogmap}} %
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| \put(6,5){\ing{lambertwmap}} %
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| \put(2,162){\sx{.7}{$y$}} %
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| \put(2,144){\sx{.6}{$6$}} %
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| \put(2,124){\sx{.6}{$4$}} %
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| \put(2,104){\sx{.6}{$2$}} %
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| \put(4,118){ \sx{.8}{\rot{-36}$v\!=\!1.8$\ero}} %
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| \put(3,96){ \sx{.8}{\rot{-20}$v\!=\!2$\ero}} %
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| \put(2, 84){\sx{.6}{$0$}} %
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| \put(8, 84){\sx{.8}{\bf cut}} %
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| \put(7,72){\sx{.7}{\rot{18}$v\!=\!-\!2$\ero}} %
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| \put(-3,64){\sx{.6}{$-2$}} %
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| \put(-3,44){\sx{.6}{$-4$}} %
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| \put(-3,24){\sx{.6}{$-6$}} %
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| \put( 22,0){\sx{.6}{$-6$}} %
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| \put( 42,0){\sx{.6}{$-4$}} %
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| \put( 62,0){\sx{.6}{$-2$}} %
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| \put( 86,0){\sx{.6}{$0$}} %
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| \put(106,0){\sx{.6}{$2$}} %
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| \put(126,0){\sx{.6}{$4$}} %
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| \put(146,0){\sx{.6}{$6$}} %
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| \put(164,0){\sx{.7}{$x$}} %
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| \put( 41, 76){\rot{-39}\sx{.8}{$u\!=\!0.8$}\ero}%
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| \put( 41, 57){\rot{-26}\sx{.84}{$u\!=\!1$}\ero}%
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| \put( 39, 40){\rot{-19}\sx{.8}{$u\!=\!1.2$}\ero}%
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| \put( 33, 21){\rot{-18}\sx{.8}{$u\!=\!1.4$}\ero}%
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| \put( 85, 145){\rot{83}\sx{.82}{$v\!=\!1$}\ero}%
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| \put(137, 102){\rot{17}\sx{.8}{$v\!=\!0.2$}\ero}%
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| \put(144, 84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}%
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| \put(140, 65){\rot{-16}\sx{.72}{$v\!=\!-\!0.2$}\ero}%
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| \put(134, 46){\rot{-32}\sx{.72}{$v\!=\!-0.4$}\ero}%
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| \put( 79, 33){\rot{-80}\sx{.86}{$v\!=\!-1$}\ero}%
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| \end{picture} %
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| } %
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| \end{document}
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| % Copyleft 2011 by Dmitrii Kouznetsov. The free use is allowed, but, please, attribute the source. It is not for any "priority", but to trace errors if any. [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]] 02:17, 26 September 2011 (UTC)
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