Difference between revisions of "File:AuTraMapT.jpg"

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[[Complex map]] of function [[AuTra]], $u\!+\!i v\!=\! \mathrm{AuTra}(x\!+\!\mathrm i y)
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Importing image file
  
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  +
[[AuTra]] is [[Abel function]] of the [[Trappmann function]] $\mathrm{tra}(z)\!=\!z\!+\!\exp(z)$.
  +
  +
==[[C++]] generator of curves==
  +
  +
// Files [[ado.cin]], [[conto.cin]], [[tania.cin]] should be loaded to the working directory in order to compile the code below
  +
  +
#include <math.h>
  +
#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 "tania.cin"
  +
#include "LambertW.cin"
  +
#include "SuZex.cin"
  +
#include "AuZex.cin"
  +
  +
z_type arctra(z_type z){return z-Tania(z-1.);}
  +
z_type tra(z_type z) {return z+exp(z);}
  +
z_type autra0(z_type z) {return auzex(exp(z));}
  +
z_type autra(z_type z) { if(fabs(Im(z))<M_PI) return auzex(exp(z));
  +
return autra(arctra(z)) +1.; }
  +
  +
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
//DB x1=-1.1259817765745026; DO(n,8){ y=Re(suzex(x1)); x=y-1.; x1+=-1.2*x; printf("%18.16f %18.16f\n", x1,y);} getchar();
  +
int M=501,M1=M+1;
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int N=1001,N1=N+1;
  +
DB X[M1],Y[N1];
  +
DB *g, *f, *w; // w is working array.
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g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
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f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
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w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
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char v[M1*N1]; // v is working array
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FILE *o;o=fopen("AuTraMap.eps","w"); ado(o,2002,2002);
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fprintf(o,"1001 1001 translate\n 100 100 scale\n");
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fprintf(o,"1 setlinejoin 2 setlinecap\n");
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DO(m,M1) X[m]=-10+.04*(m-.5);
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DO(n,N1) Y[n]=-10+.02*(n-.5);
  +
//for(n=0;n<N1;n++) { Y[n]=1.09*sinh((3./200.)*(n-200)); printf("%3d %9.6f\n",n,Y[n]); }
  +
for(m=-10;m<11;m++){M(m,-10) L(m,10) }
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for(n=-10;n<11;n++){M( -10,n) L(10,n)}
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fprintf(o,".006 W 0 0 0 RGB S\n");
  +
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]; if(m/10*10==m) printf("x=%6.3f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.019)
  +
// c=AuZex01(z-1.);
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// c=AuZexAsy(LambertW(z))+1.;
  +
//c=auzex(z);
  +
//c=suzex(c);
  +
c=autra(z);
  +
// p=abs(c-z)/(abs(c)+abs(z)); p=-log(p)/log(10.); if(p>0 && p<17) g[m*N1+n]=p;
  +
p=Re(c); q=Im(c); if(p<1000 && p>-1000 && q<1000 && q>-1000
  +
// ( x<2. || fabs(q)>1.e-12 && fabs(p)>1.e-12)
  +
){ g[m*N1+n]=p;f[m*N1+n]=q;}
  +
}}
  +
fprintf(o,"1 setlinejoin 1 setlinecap\n");
  +
p=16.;q=2.;
  +
/* p=9;q=.16;
  +
conto(o,g,w,v,X,Y,M,N,(15.3 ),-p,p); fprintf(o,".01 W .4 1 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(15. ),-p,p); fprintf(o,".02 W 1 0 1 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(14.7 ),-p,p); fprintf(o,".01 W 1 .5 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(14. ),-p,p); fprintf(o,".01 W .2 .2 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(13. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(12. ),-p,p); fprintf(o,".03 W 0 0 1 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(11. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(10. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (9. ),-p,p); fprintf(o,".03 W 0 1 1 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (8. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (7. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (6. ),-p,p); fprintf(o,".04 W 0 .5 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (5. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (4. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (3. ),-p,p); fprintf(o,".02 W 1 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (2. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (1. ),-p,p); fprintf(o,".02 W .5 0 0 RGB S\n");
  +
*/
  +
for(m=-8;m<8;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".008 W 0 .6 0 RGB S\n");
  +
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".008 W .9 0 0 RGB S\n");
  +
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".008 W 0 0 .9 RGB S\n");
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for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".032 W .8 0 0 RGB S\n");
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for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".032 W 0 0 .8 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".032 W .5 0 .5 RGB S\n");
  +
for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");
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fprintf(o,"0 setlinejoin 0 setlinecap\n");
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M(-10, M_PI)L(-1, M_PI)
  +
M(-10,-M_PI)L(-1,-M_PI) fprintf(o,"1 1 1 RGB .044 W S\n");
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DO(n,51){M(-1.-.2*n, M_PI)L(-1.-.2*(n+.4), M_PI) }
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DO(n,51){M(-1.-.2*n,-M_PI)L(-1.-.2*(n+.4),-M_PI) } fprintf(o,"0 0 0 RGB .05 W S\n");
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//#include "plofu.cin"
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fprintf(o,"showpage\n");
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fprintf(o,"%c%cTrailer\n",'%','%');
  +
fclose(o); free(f); free(g); free(w);
  +
system("epstopdf AuTraMap.eps");
  +
system( "open AuTraMap.pdf"); //for macintosh
  +
getchar(); system("killall Preview"); // For macintosh
  +
}
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==[[Latex]] generator of labels==
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%File [[AuTraMap.pdf]] should be generated with the code above in order to compile the [[Latex]] document below.
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% <nowiki> %<br>
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\documentclass[12pt]{article} % <br>
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\paperwidth 2096px % <br>
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\paperheight 2072px % <br>
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\textwidth 2394px % <br>
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\textheight 2300px % <br>
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\topmargin -101px % <br>
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\oddsidemargin -78px % <br>
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\usepackage{graphics} % <br>
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\usepackage{rotating} % <br>
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\newcommand \sx {\scalebox} % <br>
  +
\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>
  +
\parindent 0pt
  +
\pagestyle{empty}
  +
\begin{document} % <br>
  +
\newcommand \zoomax { % <br>
  +
\put(40,2048){\sx{8}{$y$}} % <br>
  +
\put(40,1870){\sx{7}{$8$}} % <br>
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\put(40,1670){\sx{7}{$6$}} % <br>
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\put(40,1470){\sx{7}{$4$}} % <br>
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\put(40,1270){\sx{7}{$2$}} % <br>
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\put(40,1070){\sx{7}{$0$}} % <br>
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\put(-12,870){\sx{7}{$-2$}} % <br>
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\put(-12,670){\sx{7}{$-4$}} % <br>
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\put(-12,470){\sx{7}{$-6$}} % <br>
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\put(-12,270){\sx{7}{$-8$}} % <br>
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\put(002, 28){\sx{7}{$-\!10$}} % <br>
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\put(220, 28){\sx{7}{$-8$}} % <br>
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\put(420, 28){\sx{7}{$-6$}} % <br>
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\put(620, 28){\sx{7}{$-4$}} % <br>
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\put(820, 28){\sx{7}{$-2$}} % <br>
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\put(1088, 28){\sx{7}{$0$}} % <br>
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\put(1288, 28){\sx{7}{$2$}} % <br>
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\put(1488, 28){\sx{7}{$4$}} % <br>
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\put(1688, 28){\sx{7}{$6$}} % <br>
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\put(1888, 28){\sx{7}{$8$}} % <br>
  +
\put(2058, 28){\sx{7}{$x$}} % <br>
  +
%\put(2166, 28){\sx{7}{$x$}} % <br>
  +
} % <br>
  +
\parindent 0pt % <br>
  +
\sx{1}{\begin{picture}(2102,2094) % <br>
  +
%\put(40,20){\ing{b271tMap3}} % <br>
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%\put(40,20){\ing{ExpMap}} % <br>
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\put(98,88){\ing{AuTraMap}} % <br>
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\zoomax % <br>
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\put(218,1384){\sx{9}{\bf cut}} % <br>
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\put(218, 752){\sx{9}{\bf cut}} % <br>
  +
%
  +
\put( 224,1836){\sx{8}{\rot{35}$v\!=\!0.6$\ero}}
  +
\put( 1590,1496){\sx{8}{\rot{70}$v\!=\!0.4$\ero}}
  +
\put( 1800,1360){\sx{8}{\rot{38}$v\!=\!0.2$\ero}}
  +
\put( 140,1068){\sx{9}{$v\!=\!0$}} % <br>
  +
\put(1874,1068){\sx{9}{$v\!=\!0$}} % <br>
  +
\put( 1740, 814){\sx{8}{\rot{-40}$v\!=\!-0.2$\ero}}
  +
\put( 194, 306){\sx{8}{\rot{-35}$v\!=\!-0.6$\ero}}
  +
%
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\put( 606, 1920){\sx{9}{\rot{-64}$u\!=\!3$\ero}}
  +
\put( 1500, 992){\sx{8.2}{\rot{90}$u\!=\!2$\ero}}
  +
\put( 1630, 960){\sx{8}{\rot{90}$u\!=\!2.2$\ero}}
  +
\put( 1846, 960){\sx{8}{\rot{90}$u\!=\!2.4$\ero}}
  +
\put( 1724, 400){\sx{8}{\rot{22}$u\!=\!2.6$\ero}}
  +
\put( 670, 290){\sx{9}{\rot{60}$u\!=\!3$\ero}}
  +
%
  +
\end{picture}} % <br>
  +
\end{document} % <br>
  +
</nowiki>
  +
  +
[[Category:AuTra]]
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[[Category:Trappmann function]]
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[[Category:Inverse function]]
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[[Category:SuZex]]
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[[Category:Tania function]]
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[[Category:Abel function]]
  +
[[Category:Regular iteration]]
  +
[[Category:Complex map]]
  +
[[Category:C++]]
  +
[[Category:Latex]]

Revision as of 09:44, 21 June 2013

Complex map of function AuTra, $u\!+\!i v\!=\! \mathrm{AuTra}(x\!+\!\mathrm i y)

AuTra is Abel function of the Trappmann function $\mathrm{tra}(z)\!=\!z\!+\!\exp(z)$.

C++ generator of curves

// Files ado.cin, conto.cin, tania.cin should be loaded to the working directory in order to compile the code below 

#include <math.h>
#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 "tania.cin"
#include "LambertW.cin"
#include "SuZex.cin"
#include "AuZex.cin"

z_type arctra(z_type z){return z-Tania(z-1.);} 
z_type tra(z_type z) {return z+exp(z);}  
z_type autra0(z_type z) {return auzex(exp(z));}
z_type autra(z_type z) { if(fabs(Im(z))<M_PI) return auzex(exp(z));
                         return autra(arctra(z)) +1.; }

main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
//DB x1=-1.1259817765745026; DO(n,8){ y=Re(suzex(x1)); x=y-1.; x1+=-1.2*x; printf("%18.16f %18.16f\n", x1,y);} getchar();
int M=501,M1=M+1;
int N=1001,N1=N+1;
DB X[M1],Y[N1];
DB *g, *f, *w; // w is working array.
g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
char v[M1*N1]; // v is working array
FILE *o;o=fopen("AuTraMap.eps","w");  ado(o,2002,2002);
fprintf(o,"1001 1001 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-10+.04*(m-.5);
DO(n,N1) Y[n]=-10+.02*(n-.5); 
//for(n=0;n<N1;n++) { Y[n]=1.09*sinh((3./200.)*(n-200)); printf("%3d %9.6f\n",n,Y[n]); }
for(m=-10;m<11;m++){M(m,-10) L(m,10)  }
for(n=-10;n<11;n++){M(  -10,n) L(10,n)}
 fprintf(o,".006 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]; if(m/10*10==m) printf("x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.019)
// c=AuZex01(z-1.);
// c=AuZexAsy(LambertW(z))+1.;
//c=auzex(z);
//c=suzex(c);
  c=autra(z);
// p=abs(c-z)/(abs(c)+abs(z)); p=-log(p)/log(10.); if(p>0 && p<17) g[m*N1+n]=p;
 p=Re(c); q=Im(c); if(p<1000 && p>-1000 && q<1000 && q>-1000 
// ( x<2. ||  fabs(q)>1.e-12 && fabs(p)>1.e-12) 
){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");
 p=16.;q=2.;
/*  p=9;q=.16;
 conto(o,g,w,v,X,Y,M,N,(15.3 ),-p,p); fprintf(o,".01 W .4 1 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(15.  ),-p,p); fprintf(o,".02 W 1 0 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14.7 ),-p,p); fprintf(o,".01 W 1 .5 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14.  ),-p,p); fprintf(o,".01 W .2 .2 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(13.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(12.  ),-p,p); fprintf(o,".03 W 0 0 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(11.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(10.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (9.  ),-p,p); fprintf(o,".03 W 0 1 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (8.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (7.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (6.  ),-p,p); fprintf(o,".04 W 0 .5 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (5.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (4.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (3.  ),-p,p); fprintf(o,".02 W 1 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (2.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (1.  ),-p,p); fprintf(o,".02 W .5 0 0 RGB S\n");
*/
for(m=-8;m<8;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".008 W 0 .6 0 RGB S\n");
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".008 W .9 0 0 RGB S\n");
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".008 W 0 0 .9 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".032 W .8 0 0 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".032 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0.  ),-2*p,2*p); fprintf(o,".032 W .5 0 .5 RGB S\n");
for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");
fprintf(o,"0 setlinejoin 0 setlinecap\n");
M(-10, M_PI)L(-1, M_PI) 
M(-10,-M_PI)L(-1,-M_PI) fprintf(o,"1 1 1 RGB .044 W S\n");
DO(n,51){M(-1.-.2*n, M_PI)L(-1.-.2*(n+.4), M_PI) }
DO(n,51){M(-1.-.2*n,-M_PI)L(-1.-.2*(n+.4),-M_PI) }   fprintf(o,"0 0 0 RGB .05 W S\n");
//#include "plofu.cin"
fprintf(o,"showpage\n");
fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o);  free(f); free(g); free(w);
      system("epstopdf AuTraMap.eps"); 
      system(    "open AuTraMap.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex generator of labels

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

% %<br> \documentclass[12pt]{article} % <br> \paperwidth 2096px % <br> \paperheight 2072px % <br> \textwidth 2394px % <br> \textheight 2300px % <br> \topmargin -101px % <br> \oddsidemargin -78px % <br> \usepackage{graphics} % <br> \usepackage{rotating} % <br> \newcommand \sx {\scalebox} % <br> \newcommand \rot {\begin{rotate}} % <br> \newcommand \ero {\end{rotate}} % <br> \newcommand \ing {\includegraphics} % <br> \newcommand \rmi {\mathrm{i}} % <br> \parindent 0pt \pagestyle{empty} \begin{document} % <br> \newcommand \zoomax { % <br> \put(40,2048){\sx{8}{$y$}} % <br> \put(40,1870){\sx{7}{$8$}} % <br> \put(40,1670){\sx{7}{$6$}} % <br> \put(40,1470){\sx{7}{$4$}} % <br> \put(40,1270){\sx{7}{$2$}} % <br> \put(40,1070){\sx{7}{$0$}} % <br> \put(-12,870){\sx{7}{$-2$}} % <br> \put(-12,670){\sx{7}{$-4$}} % <br> \put(-12,470){\sx{7}{$-6$}} % <br> \put(-12,270){\sx{7}{$-8$}} % <br> \put(002, 28){\sx{7}{$-\!10$}} % <br> \put(220, 28){\sx{7}{$-8$}} % <br> \put(420, 28){\sx{7}{$-6$}} % <br> \put(620, 28){\sx{7}{$-4$}} % <br> \put(820, 28){\sx{7}{$-2$}} % <br> \put(1088, 28){\sx{7}{$0$}} % <br> \put(1288, 28){\sx{7}{$2$}} % <br> \put(1488, 28){\sx{7}{$4$}} % <br> \put(1688, 28){\sx{7}{$6$}} % <br> \put(1888, 28){\sx{7}{$8$}} % <br> \put(2058, 28){\sx{7}{$x$}} % <br> %\put(2166, 28){\sx{7}{$x$}} % <br> } % <br> \parindent 0pt % <br> \sx{1}{\begin{picture}(2102,2094) % <br> %\put(40,20){\ing{b271tMap3}} % <br> %\put(40,20){\ing{ExpMap}} % <br> \put(98,88){\ing{AuTraMap}} % <br> \zoomax % <br> \put(218,1384){\sx{9}{\bf cut}} % <br> \put(218, 752){\sx{9}{\bf cut}} % <br> % \put( 224,1836){\sx{8}{\rot{35}$v\!=\!0.6$\ero}} \put( 1590,1496){\sx{8}{\rot{70}$v\!=\!0.4$\ero}} \put( 1800,1360){\sx{8}{\rot{38}$v\!=\!0.2$\ero}} \put( 140,1068){\sx{9}{$v\!=\!0$}} % <br> \put(1874,1068){\sx{9}{$v\!=\!0$}} % <br> \put( 1740, 814){\sx{8}{\rot{-40}$v\!=\!-0.2$\ero}} \put( 194, 306){\sx{8}{\rot{-35}$v\!=\!-0.6$\ero}} % \put( 606, 1920){\sx{9}{\rot{-64}$u\!=\!3$\ero}} \put( 1500, 992){\sx{8.2}{\rot{90}$u\!=\!2$\ero}} \put( 1630, 960){\sx{8}{\rot{90}$u\!=\!2.2$\ero}} \put( 1846, 960){\sx{8}{\rot{90}$u\!=\!2.4$\ero}} \put( 1724, 400){\sx{8}{\rot{22}$u\!=\!2.6$\ero}} \put( 670, 290){\sx{9}{\rot{60}$u\!=\!3$\ero}} % \end{picture}} % <br> \end{document} % <br>

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current17:50, 20 June 2013Thumbnail for version as of 17:50, 20 June 20134,367 × 4,317 (1.82 MB)Maintenance script (talk | contribs)Importing image file

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