Difference between revisions of "File:Ack4d.jpg"

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[[Complex map]] of [[tetration]] to base 10.
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Importing image file
  
  +
  +
$u+\mathrm i v=\mathrm{tet}_{10}(x\!+\!\mathrm i y)$
  +
  +
Prepared as Figure 4d for publication in Journal [[Applied and Computational Mathematics]], but not used in the final version
  +
<ref>http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf
  +
http://mizugadro.mydns.jp/PAPERS/2014acker.pdf
  +
D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.
  +
</ref>
  +
  +
==References==
  +
<references/>
  +
  +
==[[C++]] generator of map==
  +
  +
Files
  +
[[ado.cin]],
  +
[[conto.cin]],
  +
[[filog.cin]],
  +
[[f4ten.cin]],
  +
[[GLxw2048.inc]],
  +
[[f2048ten.inc]]
  +
should be loaded in order to compile the code below
  +
  +
<poem><nomathjax><nowiki>
  +
#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 std::complex<double> z_type;
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
#include "conto.cin"
  +
#include "filog.cin"
  +
  +
//z_type b=z_type( 1.5259833851700000, 0.0178411853321000);
  +
z_type b=10.;
  +
z_type a=log(b);
  +
z_type Zo=Filog(a);
  +
z_type Zc=conj(Filog(conj(a)));
  +
DB A=32.;
  +
  +
/*
  +
z_type tetb(z_type z){ int k; DB t; z_type c, cu,cd;
  +
#include "GLxw2048.inc"
  +
int K=2048;
  +
//#include "ima6.inc"
  +
#include "TetSheldonIma.inc"
  +
z_type E[2048],G[2048];
  +
DO(k,K){c=F[k]; E[k]=log(c)/a; G[k]=exp(a*c);}
  +
c=0.;
  +
z+=z_type(0.1196573712872846, 0.1299776198056910);
  +
DO(k,K){t=A*GLx[k];c+=GLw[k]*(G[k]/(z_type( 1.,t)-z)-E[k]/(z_type(-1.,t)-z));}
  +
cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) );
  +
cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) );
  +
c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd;
  +
return c;}
  +
*/
  +
  +
  +
z_type f4ten(z_type z){ //NOT SHIFTED FOR x1 !!!!
  +
#include "GLxw2048.inc"
  +
z_type tenZo=z_type(-0.119193073414549, 0.750583293932439);
  +
z_type tenZc=z_type(-0.119193073414549,-0.750583293932439);
  +
  +
DB Lten= 2.302585092994046;
  +
z_type tenQ=z_type( 0.559580251215472, 1.728281903659204);// =L*Zo+log(L)
  +
z_type tenT=z_type( 3.290552906607012, 1.065409768058325);
  +
// #include "tenzo.inc"
  +
#include"f2048ten.inc"
  +
//Aten is defined there
  +
int j,k,m,n; DB x,y, u, t; z_type c,d, cu,cd;
  +
// z_type E[K],G[K];
  +
z_type E[2048],G[2048];
  +
DO(k,K){c=F[k];E[k]=log(c)/Lten;G[k]=exp(c*Lten);}
  +
// the initioalization abouve should run at the compillation
  +
c=0.;
  +
DO(k,K){t=Aten*GLx[k]; c+= GLw[k]*( G[k]/(z_type( 1.,t)-z) - E[k]/(z_type(-1.,t)-z) );}
  +
cu=.5-I/(2.*M_PI)*log( (z_type(1.,-Aten)+z)/(z_type(1., Aten)-z) );
  +
cd=.5-I/(2.*M_PI)*log( (z_type(1.,-Aten)-z)/(z_type(1., Aten)+z) );
  +
return c*(Aten/(2.*M_PI)) +tenZo*cu+tenZc*cd;
  +
}
  +
DB x1ten= 0.0377406857309657;
  +
//#include "figx1ten.inc"
  +
DB Lten=log(10.);
  +
z_type F4TEN(z_type z){ DB x=Re(z);
  +
// DB L=log(2.);
  +
if(x<-.5) return log(F4TEN(z+1.))/Lten;
  +
if(x> .5) return exp(F4TEN(z-1.)*Lten);
  +
return f4ten(z+x1ten);
  +
}
  +
// f4ten end
  +
  +
int main(){ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d;
  +
//int M=161,M1=M+1;
  +
int M=601,M1=M+1;
  +
int N=461,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("10.eps","w");ado(o,602,202);
  +
fprintf(o,"301 101 translate\n 10 10 scale\n");
  +
DO(m,M1)X[m]=-30.+.1*(m);
  +
DO(n,200)Y[n]=-10.+.05*n;
  +
Y[200]=-.01;
  +
Y[201]= .01;
  +
for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-1.);
  +
for(m=-30;m<31;m++){if(m==0){M(m,-10.2)L(m,10.2)} else{M(m,-10)L(m,10)}}
  +
for(n=-10;n<11;n++){ M( -30,n)L(30,n)}
  +
fprintf(o,".008 W 0 0 0 RGB S\n");
  +
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
  +
  +
DO(n,N1){y=Y[n];
  +
for(m=295;m<305;m++)
  +
{x=X[m]; //printf("%5.2f\n",x);
  +
z=z_type(x,y);
  +
// c=tetb(z);
  +
c=F4TEN(z);
  +
p=Re(c);q=Im(c);
  +
if(p>-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
  +
d=c;
  +
for(k=1;k<31;k++)
  +
{ m1=m+k*10; if(m1>M) break;
  +
d=exp(a*d);
  +
p=Re(d);q=Im(d);
  +
if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
  +
}
  +
d=c;
  +
for(k=1;k<31;k++)
  +
{ m1=m-k*10; if(m1<0) break;
  +
d=log(d)/a;
  +
p=Re(d);q=Im(d);
  +
if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
  +
}
  +
}}
  +
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1;q=.5;
  +
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,".02 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,".02 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,".02 W 0 0 .9 RGB S\n");
  +
for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".08 W .9 0 0 RGB S\n");
  +
for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 .9 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".08 W .6 0 .6 RGB S\n");
  +
for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 0 RGB S\n");
  +
// y= 0; for(m=0;m<260;m+=6) {x=-2.-.1*m; M(x,y) L(x-.1,y)}
  +
// fprintf(o,".07 W 1 .5 0 RGB S\n");
  +
// y= 0; for(m=3;m<260;m+=6) {x=-2-.1*m; M(x,y) L(x-.1,y)}
  +
// fprintf(o,".07 W 0 .5 1 RGB S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf 10.eps");
  +
system( "open 10.pdf");
  +
getchar(); system("killall Preview");
  +
}
  +
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
  +
<poem><nomathjax><nowiki>
  +
\documentclass{amsproc}
  +
\usepackage{graphicx}
  +
\usepackage{rotating}
  +
\usepackage{hyperref}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rme {{\rm e}} %%makes the base of natural logarithms Roman font
  +
%\newcommand \rme {{e}} %%makes the base of natural logarithms Italics font; choose one of these
  +
\newcommand \rmi {{\rm i}} %%imaginary unity is always roman font
  +
\newcommand \ds {\displaystyle}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing \includegraphics
  +
\usepackage{geometry}
  +
\topmargin -94pt
  +
%\topmargin -97pt
  +
\oddsidemargin -87pt
  +
\paperwidth 618pt
  +
%\paperheight 216pt
  +
\paperheight 212pt
  +
\begin{document}
  +
  +
\newcommand \mapax {
  +
\put(2,206){\sx{1.2}{$y$}}
  +
\put(2,188){\sx{1.2}{$8$}}
  +
\put(2,168){\sx{1.2}{$6$}}
  +
\put(2,148){\sx{1.2}{$4$}}
  +
\put(2,128){\sx{1.2}{$2$}}
  +
\put(2,108){\sx{1.2}{$0$}}
  +
\put(-6,88){\sx{1.2}{$-2$}}
  +
\put(-6,68){\sx{1.2}{$-4$}}
  +
\put(-6,48){\sx{1.2}{$-6$}}
  +
\put(-6,28){\sx{1.2}{$-8$}}
  +
\put(-1,1){\sx{1.2}{$-30$}}
  +
\put( 49,1){\sx{1.2}{$-25$}}
  +
\put( 99,1){\sx{1.2}{$-20$}}
  +
\put(149,1){\sx{1.2}{$-15$}}
  +
\put(199,1){\sx{1.2}{$-10$}}
  +
\put(252,1){\sx{1.2}{$-5$}}
  +
\put(309,1){\sx{1.2}{$0$}}
  +
\put(329,1){\sx{1.2}{$2$}}
  +
\put(349,1){\sx{1.2}{$4$}}
  +
\put(369,1){\sx{1.2}{$6$}}
  +
\put(389,1){\sx{1.2}{$8$}}
  +
\put(407,1){\sx{1.2}{$10$}}
  +
\put(457,1){\sx{1.2}{$15$}}
  +
\put(507,1){\sx{1.2}{$20$}}
  +
\put(557,1){\sx{1.2}{$25$}}
  +
\put(607,1){\sx{1.2}{$x$}}
  +
}
  +
%\flushright{$b=\mathrm e \approx 2.71$}
  +
%\sx{.586}
  +
{\begin{picture}(620,212) %%%%%%%%%%%%%%%%%%%%%%
  +
%\put(10,10){\ing{tetema}} \mapax
  +
\put(10,10){\ing{10}} \mapax
  +
  +
%\multiput(110,118)(56.1,10.7){8}{\sx{1.2}{$u\!=\!0.8$}}
  +
%\multiput(256,120)(56.1,10.7){7}{\sx{1.2}{\rot{20}$u\!=\!1$\ero}}
  +
%\multiput(302,120)(56.1,10.2){7}{\sx{1.2}{\rot{0}$v\!=\!1$\ero}}
  +
  +
%\multiput(189,116)(44.7,10.5){9}{\sx{1.2}{$u\!=\!1.4$}}
  +
\multiput(256,124)(33.2,10.7){8}{\sx{1.2}{$u\!=\!-1.2$}}
  +
\put(25,108.4){\sx{1.4}{\bf cut}} \put(296,108.4){\sx{1.2}{$v\!=\!0$}}
  +
\multiput(242,92)(33.2,-10.7){8}{\sx{1.2}{$v\!=\!-0.8$}}
  +
%\multiput(193,93)(44.7,-10.5){8}{\sx{1.2}{$v\!=\!-1.4$}}
  +
  +
%\put(20,200){\sx{1.4}{$u+\mathrm i v \approx 0.31813150520 + 1.3372357014 \,\mathrm i$}}
  +
%\put(30, 20){\sx{1.4}{$u+\mathrm i v \approx 0.31813150520 - 1.3372357014 \,\mathrm i$}}
  +
\put(20,196){\sx{1.4}{$u+\mathrm i v \approx -0.11919307341+ 0.75058329393 \,\mathrm i$}}
  +
\put(30, 20){\sx{1.4}{$u+\mathrm i v \approx -0.11919307341- 0.75058329393 \,\mathrm i$}}
  +
\end{picture}}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
[[Category:Applied and Computational Mathematics]]
  +
[[Category:AMS]]
  +
[[Category:Base 10]]
  +
[[Category:C++]]
  +
[[Category:Complex map]]
  +
[[Category:Latex]]
  +
[[Category:Tetration]]

Latest revision as of 08:28, 1 December 2018

Complex map of tetration to base 10.

$u+\mathrm i v=\mathrm{tet}_{10}(x\!+\!\mathrm i y)$

Prepared as Figure 4d for publication in Journal Applied and Computational Mathematics, but not used in the final version [1]

References

  1. ↑ http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf http://mizugadro.mydns.jp/PAPERS/2014acker.pdf D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.

C++ generator of map

Files ado.cin, conto.cin, filog.cin, f4ten.cin, GLxw2048.inc, f2048ten.inc should be loaded 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 std::complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "conto.cin"
 #include "filog.cin"

//z_type b=z_type( 1.5259833851700000, 0.0178411853321000);
z_type b=10.;
z_type a=log(b);
z_type Zo=Filog(a);
z_type Zc=conj(Filog(conj(a)));
DB A=32.;

/*
z_type tetb(z_type z){ int k; DB t; z_type c, cu,cd;
#include "GLxw2048.inc"
int K=2048;
//#include "ima6.inc"
#include "TetSheldonIma.inc"
z_type E[2048],G[2048];
DO(k,K){c=F[k]; E[k]=log(c)/a; G[k]=exp(a*c);}
c=0.;
z+=z_type(0.1196573712872846, 0.1299776198056910);
DO(k,K){t=A*GLx[k];c+=GLw[k]*(G[k]/(z_type( 1.,t)-z)-E[k]/(z_type(-1.,t)-z));}
 cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) );
 cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) );
 c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd;
 return c;}
*/


z_type f4ten(z_type z){ //NOT SHIFTED FOR x1 !!!!
#include "GLxw2048.inc"
z_type tenZo=z_type(-0.119193073414549, 0.750583293932439);
z_type tenZc=z_type(-0.119193073414549,-0.750583293932439);

DB Lten= 2.302585092994046;
z_type tenQ=z_type( 0.559580251215472, 1.728281903659204);// =L*Zo+log(L)
z_type tenT=z_type( 3.290552906607012, 1.065409768058325);
// #include "tenzo.inc"
       #include"f2048ten.inc"
       //Aten is defined there
        int j,k,m,n; DB x,y, u, t; z_type c,d, cu,cd;
// z_type E[K],G[K];
       z_type E[2048],G[2048];
       DO(k,K){c=F[k];E[k]=log(c)/Lten;G[k]=exp(c*Lten);}
// the initioalization abouve should run at the compillation
       c=0.;
       DO(k,K){t=Aten*GLx[k]; c+= GLw[k]*( G[k]/(z_type( 1.,t)-z) - E[k]/(z_type(-1.,t)-z) );}
       cu=.5-I/(2.*M_PI)*log( (z_type(1.,-Aten)+z)/(z_type(1., Aten)-z) );
       cd=.5-I/(2.*M_PI)*log( (z_type(1.,-Aten)-z)/(z_type(1., Aten)+z) );
       return c*(Aten/(2.*M_PI)) +tenZo*cu+tenZc*cd;
}
DB x1ten= 0.0377406857309657;
//#include "figx1ten.inc"
DB Lten=log(10.);
z_type F4TEN(z_type z){ DB x=Re(z);
// DB L=log(2.);
                       if(x<-.5) return log(F4TEN(z+1.))/Lten;
                       if(x> .5) return exp(F4TEN(z-1.)*Lten);
                       return f4ten(z+x1ten);
               }
// f4ten end

int main(){ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d;
 //int M=161,M1=M+1;
 int M=601,M1=M+1;
 int N=461,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("10.eps","w");ado(o,602,202);
 fprintf(o,"301 101 translate\n 10 10 scale\n");
 DO(m,M1)X[m]=-30.+.1*(m);
 DO(n,200)Y[n]=-10.+.05*n;
         Y[200]=-.01;
         Y[201]= .01;
 for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-1.);
 for(m=-30;m<31;m++){if(m==0){M(m,-10.2)L(m,10.2)} else{M(m,-10)L(m,10)}}
 for(n=-10;n<11;n++){ M( -30,n)L(30,n)}
 fprintf(o,".008 W 0 0 0 RGB S\n");
 DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}

 DO(n,N1){y=Y[n];
          for(m=295;m<305;m++)
          {x=X[m]; //printf("%5.2f\n",x);
           z=z_type(x,y);
// c=tetb(z);
           c=F4TEN(z);
           p=Re(c);q=Im(c);
           if(p>-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
           d=c;
           for(k=1;k<31;k++)
                { m1=m+k*10; if(m1>M) break;
                d=exp(a*d);
                p=Re(d);q=Im(d);
                if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
                }
           d=c;
           for(k=1;k<31;k++)
                { m1=m-k*10; if(m1<0) break;
                d=log(d)/a;
                p=Re(d);q=Im(d);
                if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
                }
        }}

 fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1;q=.5;
 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,".02 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,".02 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,".02 W 0 0 .9 RGB S\n");
 for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".08 W .9 0 0 RGB S\n");
 for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 .9 RGB S\n");
                    conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".08 W .6 0 .6 RGB S\n");
 for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 0 RGB S\n");
// y= 0; for(m=0;m<260;m+=6) {x=-2.-.1*m; M(x,y) L(x-.1,y)}
// fprintf(o,".07 W 1 .5 0 RGB S\n");
// y= 0; for(m=3;m<260;m+=6) {x=-2-.1*m; M(x,y) L(x-.1,y)}
// fprintf(o,".07 W 0 .5 1 RGB S\n");
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf 10.eps");
        system( "open 10.pdf");
        getchar(); system("killall Preview");
 }

Latex generator of labels


\documentclass{amsproc}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{hyperref}
\newcommand \sx {\scalebox}
\newcommand \rme {{\rm e}} %%makes the base of natural logarithms Roman font
%\newcommand \rme {{e}} %%makes the base of natural logarithms Italics font; choose one of these
\newcommand \rmi {{\rm i}} %%imaginary unity is always roman font
\newcommand \ds {\displaystyle}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing \includegraphics
\usepackage{geometry}
\topmargin -94pt
%\topmargin -97pt
\oddsidemargin -87pt
\paperwidth 618pt
%\paperheight 216pt
\paperheight 212pt
\begin{document}

\newcommand \mapax {
\put(2,206){\sx{1.2}{$y$}}
\put(2,188){\sx{1.2}{$8$}}
\put(2,168){\sx{1.2}{$6$}}
\put(2,148){\sx{1.2}{$4$}}
\put(2,128){\sx{1.2}{$2$}}
\put(2,108){\sx{1.2}{$0$}}
\put(-6,88){\sx{1.2}{$-2$}}
\put(-6,68){\sx{1.2}{$-4$}}
\put(-6,48){\sx{1.2}{$-6$}}
\put(-6,28){\sx{1.2}{$-8$}}
\put(-1,1){\sx{1.2}{$-30$}}
\put( 49,1){\sx{1.2}{$-25$}}
\put( 99,1){\sx{1.2}{$-20$}}
\put(149,1){\sx{1.2}{$-15$}}
\put(199,1){\sx{1.2}{$-10$}}
\put(252,1){\sx{1.2}{$-5$}}
\put(309,1){\sx{1.2}{$0$}}
\put(329,1){\sx{1.2}{$2$}}
\put(349,1){\sx{1.2}{$4$}}
\put(369,1){\sx{1.2}{$6$}}
\put(389,1){\sx{1.2}{$8$}}
\put(407,1){\sx{1.2}{$10$}}
\put(457,1){\sx{1.2}{$15$}}
\put(507,1){\sx{1.2}{$20$}}
\put(557,1){\sx{1.2}{$25$}}
\put(607,1){\sx{1.2}{$x$}}
}
%\flushright{$b=\mathrm e \approx 2.71$}
%\sx{.586}
{\begin{picture}(620,212) %%%%%%%%%%%%%%%%%%%%%%
%\put(10,10){\ing{tetema}} \mapax
\put(10,10){\ing{10}} \mapax

%\multiput(110,118)(56.1,10.7){8}{\sx{1.2}{$u\!=\!0.8$}}
%\multiput(256,120)(56.1,10.7){7}{\sx{1.2}{\rot{20}$u\!=\!1$\ero}}
%\multiput(302,120)(56.1,10.2){7}{\sx{1.2}{\rot{0}$v\!=\!1$\ero}}

%\multiput(189,116)(44.7,10.5){9}{\sx{1.2}{$u\!=\!1.4$}}
\multiput(256,124)(33.2,10.7){8}{\sx{1.2}{$u\!=\!-1.2$}}
\put(25,108.4){\sx{1.4}{\bf cut}} \put(296,108.4){\sx{1.2}{$v\!=\!0$}}
\multiput(242,92)(33.2,-10.7){8}{\sx{1.2}{$v\!=\!-0.8$}}
%\multiput(193,93)(44.7,-10.5){8}{\sx{1.2}{$v\!=\!-1.4$}}

%\put(20,200){\sx{1.4}{$u+\mathrm i v \approx 0.31813150520 + 1.3372357014 \,\mathrm i$}}
%\put(30, 20){\sx{1.4}{$u+\mathrm i v \approx 0.31813150520 - 1.3372357014 \,\mathrm i$}}
\put(20,196){\sx{1.4}{$u+\mathrm i v \approx -0.11919307341+ 0.75058329393 \,\mathrm i$}}
\put(30, 20){\sx{1.4}{$u+\mathrm i v \approx -0.11919307341- 0.75058329393 \,\mathrm i$}}
\end{picture}}
\end{document}

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