Difference between pages "File:Vladi07.jpg" and "File:Vladi08.jpg"

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Line 1: Line 1:
Maps of agreement of approximations of natural [[tetration]] with elementary functions [[fima]] and [[maclo]], used in the implementation [[fsexp.cin]].
+
Agreement of approximations of tetration by elementary functions with the original representation through the Cauchi integral is shown in the left hand side map.
   
  +
The similar map at the right hand side represents the comparison to the similar representation through the Cauchi integral, but the contour of integration is displaced for 1/2 to the left.
Left: $D=D_6(x\!+\!\mathrm i y)$;
 
  +
  +
Levels $D=\mathrm{const}$ are drawn with step 2, but the exception is dome for level $D=1$, this level is drawn with thick lines.
   
 
$\displaystyle
 
$\displaystyle
  +
{\rm fse}(z)=\left\{ \begin{array}{ccrcccr}
D_6(z)= - \ln \left( \frac
 
{|\mathrm{tai}(z) - \mathrm{fima}(z)|}
+
{\rm fima}(z) &,& 4.5& \!<\! &\Im(z) \\
{|\mathrm{tai}(z)|+|\mathrm{fima}(z)|}
+
{\rm tai}(z) &,& 1.5& \!<\! &\Im(z) &\!\le\!& 4.5 \\
  +
{\rm maclo}(z) &,& -1.5& \le &\Im(z) &\!\le\!& 1.5 \\
\right)$
 
  +
{\rm tai}(z^{*})^{*} &,& -4.5& \le &\Im(z) &\!<\!&\!\! -1.5 \\
  +
{\rm fima}(z^{*})^{*} &,& & &\Im(z) &\!<\!&\!\! -4.5
  +
\end{array}
  +
\right.$
   
  +
Mnemonics for the name of the approximation: Fast Super Exponent.
right: $D=D_7(x\!+\!\mathrm i y)$;
 
  +
  +
The agreement plotted is
   
 
$\displaystyle
 
$\displaystyle
D_7(z)= - \ln \left( \frac
+
D=D_{8}(z)=-\lg\left( \frac
{|\mathrm{tai}(z) - \mathrm{maclo}(z)|}
+
{|\mathrm{fse}(z)-F_{4}(z)|}
{|\mathrm{tai}(z)|+|\mathrm{maclo}(z)|}
+
{|\mathrm{fse}(z)|+|F_{4}(z)|}
\right)$
+
\right)
  +
$
   
  +
where $F_4$ denotes the 4th [[ackermann]], id set, natural [[tetration]] evaluated through the Cauchi integral
Levels $D=\mathrm{const}$ are drawn with step 2, but the exception is dome for level $D=1$, this level is drawn with thick lines.
 
  +
<ref>
  +
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html <br>
  +
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf <br>
  +
http://mizugadro.mudns.jp/PAPERS/2009analuxpRepri.pdf
  +
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
  +
</ref>.
   
Usage: this is figure 14.10 of the book [[Суперфункции]] (2014, In Russian) <ref>
+
Usage: this is figure 14.11 of the book [[Суперфункции]] (2014, In Russian) <ref>
 
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
 
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
 
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf <br>
 
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf <br>
Line 30: Line 45:
 
http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf
 
http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf
 
D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
 
D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
Figure 7.
+
Figure 8.
 
</ref>.
 
</ref>.
   
Line 44: Line 59:
   
 
<poem><nomathjax><nowiki>
 
<poem><nomathjax><nowiki>
  +
 
#include <math.h>
 
#include <math.h>
 
#include <stdio.h>
 
#include <stdio.h>
Line 51: Line 67:
 
#include <complex>
 
#include <complex>
 
typedef std::complex<double> z_type;
 
typedef std::complex<double> z_type;
//#include <complex.h>
 
//#define z_type complex<double>
 
 
#define Re(x) x.real()
 
#define Re(x) x.real()
 
#define Im(x) x.imag()
 
#define Im(x) x.imag()
 
#define I z_type(0.,1.)
 
#define I z_type(0.,1.)
   
  +
//#include "superex.cin"
 
#include "fsexp.cin"
 
#include "fsexp.cin"
  +
#include "f4natu.cin"
  +
//z_type FSEXP(z_type z){DB y=Im(z);
   
  +
z_type fsexp(z_type z){DB y=Im(z);
  +
if(y> 4.5) return fima(z);
  +
if(y> 1.5) return tai3(z);
  +
if(y>-1.5) return maclo(z);
  +
if(y>-4.5) return conj(tai3(conj(z)));
  +
return conj(fima(conj(z)));
  +
}
  +
//#include "superlo.cin"
 
#include "conto.cin"
 
#include "conto.cin"
 
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
 
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
z_type Zo=z_type(.31813150520476413, 1.3372357014306895);
  +
z_type Zc=z_type(.31813150520476413,-1.3372357014306895);
   
 
int M=100,M1=M+1;
 
int M=100,M1=M+1;
Line 66: Line 93:
 
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
 
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
 
char v[M1*N1]; // v is working array
FILE *o;o=fopen("vladi07a.eps","w");ado(o,82,82);
+
//FILE *o;o=fopen("figcf4small.eps","w");ado(o,0,0,82,82);
  +
FILE *o;o=fopen("vladi08a.eps","w");ado(o,82,82);
 
fprintf(o,"41 11 translate\n 10 10 scale\n");
 
fprintf(o,"41 11 translate\n 10 10 scale\n");
   
 
DO(m,M1) X[m]=-4.+.08*(m-.5);
 
DO(m,M1) X[m]=-4.+.08*(m-.5);
DO(n,N1)Y[n]= -1 +.08*(n-.5);
+
DO(n,N1) Y[n]=-1.+.08*(n-.5);
   
 
for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
 
for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
Line 79: Line 107:
 
DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x);
 
DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x);
 
DO(n,N1){y=Y[n]; z=z_type(x,y);
 
DO(n,N1){y=Y[n]; z=z_type(x,y);
c=tai3(z);
+
c=fsexp(z);
// d=fima1(z);
+
d=F4natu(z);
d=fima(z);
+
p=abs(c-d);///(abs(c)+abs(d));
p=abs(c-d)/(abs(c)+abs(d));
 
 
p=-log(p)/log(10.);
 
p=-log(p)/log(10.);
// p=Re(c); q=Im(c);
+
// p=Re(c); q=Im(c);
 
if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
 
if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
 
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
 
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
 
}}
 
}}
 
 
#include"plodi.cin"
 
#include"plodi.cin"
  +
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n");
 
 
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
 
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
system("epstopdf vladi07a.eps");
+
// system( "gv fig08a.eps");
system( "open vladi07a.pdf");//macintosh
+
system("epstopdf vladi08a.eps");
// getchar(); system("killall Preview"); //macintosh
+
system( "open vladi08a.pdf"); // for linux
  +
// getchar(); system("killall Preview");// for macintosh
 
}
 
}
  +
 
</nowiki></nomathjax></poem>
 
</nowiki></nomathjax></poem>
   
 
==[[C++]] generator of the second picture==
 
==[[C++]] generator of the second picture==
  +
  +
[[vladif5c.cin]] also should be loaded
  +
 
<poem><nomathjax><nowiki>
 
<poem><nomathjax><nowiki>
   
Line 108: Line 139:
 
#include <complex>
 
#include <complex>
 
typedef std::complex<double> z_type;
 
typedef std::complex<double> z_type;
//#include <complex.h>
 
//#define z_type complex<double>
 
 
#define Re(x) x.real()
 
#define Re(x) x.real()
 
#define Im(x) x.imag()
 
#define Im(x) x.imag()
 
#define I z_type(0.,1.)
 
#define I z_type(0.,1.)
   
  +
//#include "superex.cin"
 
#include "fsexp.cin"
 
#include "fsexp.cin"
  +
//#include "f4natu.cin"
  +
#include "vladif5c.cin"
  +
//z_type FSEXP(z_type z){DB y=Im(z);
   
  +
z_type fsexp(z_type z){DB y=Im(z);
  +
if(y> 4.5) return fima(z);
  +
if(y> 1.5) return tai3(z);
  +
if(y>-1.5) return maclo(z);
  +
if(y>-4.5) return conj(tai3(conj(z)));
  +
return conj(fima(conj(z)));
  +
}
  +
//#include "superlo.cin"
 
#include "conto.cin"
 
#include "conto.cin"
 
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
 
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
z_type Zo=z_type(.31813150520476413, 1.3372357014306895);
  +
z_type Zc=z_type(.31813150520476413,-1.3372357014306895);
   
 
int M=100,M1=M+1;
 
int M=100,M1=M+1;
Line 123: Line 166:
 
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
 
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
 
char v[M1*N1]; // v is working array
FILE *o;o=fopen("vladi07b.eps","w");ado(o,82,82);
+
//FILE *o;o=fopen("figcf5small.eps","w");ado(o,0,0,82,82);
  +
FILE *o;o=fopen("vladi08b.eps","w");ado(o,82,82);
 
fprintf(o,"41 11 translate\n 10 10 scale\n");
 
fprintf(o,"41 11 translate\n 10 10 scale\n");
   
DO(m,M1) X[m]=-4.+.08*(m-.5);
+
DO(m,M1)X[m]=-4.+.08*(m-.5);
DO(n,N1)Y[n]= -1 +.08*(n-.5);
+
DO(n,N1)Y[n]=-1.+.08*(n-.5);
   
 
for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
 
for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
for(n=0;n<7;n++) { M( -3,n)L(3,n)}
+
for(n= 0;n<7;n++) { M( -3,n)L(3,n)}
 
fprintf(o,".006 W 0 0 0 RGB S\n");
 
fprintf(o,".006 W 0 0 0 RGB S\n");
   
Line 136: Line 180:
 
DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x);
 
DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x);
 
DO(n,N1){y=Y[n]; z=z_type(x,y);
 
DO(n,N1){y=Y[n]; z=z_type(x,y);
c=tai3(z);
+
c=fsexp(z);
d=maclo(z);
+
d=F5(z);
// d=fima1(z);
+
p=abs(c-d);///(abs(c)+abs(d));
p=abs(c-d)/(abs(c)+abs(d));
 
 
p=-log(p)/log(10.);
 
p=-log(p)/log(10.);
// p=Re(c); q=Im(c);
+
// p=Re(c); q=Im(c);
if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
+
if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
+
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
}}
+
}}
   
 
#include"plodi.cin"
 
#include"plodi.cin"
  +
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n");
 
 
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
 
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
system("epstopdf vladi07b.eps");
+
system("epstopdf vladi08b.eps");
system( "open vladi07b.pdf");//macintosh
+
system( "open vladi08b.pdf");//linux
// getchar(); system("killall Preview");//macintosh
+
//getchar(); system("killall Preview");
 
}
 
}
 
 
</nowiki></nomathjax></poem>
 
</nowiki></nomathjax></poem>
   
Line 162: Line 204:
 
\usepackage{rotating}
 
\usepackage{rotating}
 
\usepackage{geometry}
 
\usepackage{geometry}
\paperwidth 356px
+
\paperwidth 378px
 
%\paperheight 134px
 
%\paperheight 134px
\paperheight 184px
+
\paperheight 180px
\topmargin -98pt
+
\topmargin -104pt
 
\oddsidemargin -94pt
 
\oddsidemargin -94pt
 
\pagestyle{empty}
 
\pagestyle{empty}
Line 175: Line 217:
 
\newcommand \ero {\end{rotate}}
 
\newcommand \ero {\end{rotate}}
   
  +
\sx{2.2}{\begin{picture}(90,80)
\newcommand \tafiax {
 
\put(7,68){\sx{.5}{$y$}}
+
%\put(0,0){\includegraphics{figsexpF4}}
%\put(5,74){\sx{.5}{$\Im(z)$}}
+
%\put(0,0){\includegraphics{figcf4small}}
%\put(7,69){\sx{.5}{$6$}}
+
\put(0,0){\includegraphics{vladi08a}}
  +
\put(5,68){\sx{.45}{$\Im(z)$}}
 
\put(7,59){\sx{.5}{$5$}}
 
\put(7,59){\sx{.5}{$5$}}
 
\put(7,49){\sx{.5}{$4$}}
 
\put(7,49){\sx{.5}{$4$}}
Line 185: Line 228:
 
\put(7,19){\sx{.5}{$1$}}
 
\put(7,19){\sx{.5}{$1$}}
 
\put(7, 9){\sx{.5}{$0$}}
 
\put(7, 9){\sx{.5}{$0$}}
\put( 7 ,6){\sx{.5}{$-\!3$}}
+
\put(70 , 6){\sx{.45}{$\Re(z)$}}
\put(17 ,6){\sx{.5}{$-\!2$}}
+
\put(60 , 6){\sx{.5}{$2$}}
\put(27 ,6){\sx{.5}{$-\!1$}}
 
 
\put(40 , 6){\sx{.5}{$0$}}
 
\put(40 , 6){\sx{.5}{$0$}}
\put(50 , 6){\sx{.5}{$1$}}
+
\put(17 ,6){\sx{.5}{$-\!2$}}
  +
\put(25,62){\sx{.5}{$D\!>\!14$}}
  +
\put(55,60){\sx{.5}{$12\!<\!D\!<\!14$}}
  +
\put(68,20){\sx{.5}{$D\!<\!1$}}
  +
\end{picture}}
  +
\sx{2.2}{\begin{picture}(80,80)
  +
%\put(0,0){\includegraphics{figsexpF5}}
  +
%\put(0,0){\includegraphics{figcf5small}}
  +
\put(0,0){\includegraphics{vladi08b}}
  +
\put(5,68){\sx{.45}{$\Im(z)$}}
  +
\put(7,59){\sx{.5}{$5$}}
  +
\put(7,49){\sx{.5}{$4$}}
  +
\put(7,39){\sx{.5}{$3$}}
  +
\put(7,29){\sx{.5}{$2$}}
  +
\put(7,19){\sx{.5}{$1$}}
  +
\put(7, 9){\sx{.5}{$0$}}
  +
\put(70 , 6){\sx{.45}{$\Re(z)$}}
 
\put(60 , 6){\sx{.5}{$2$}}
 
\put(60 , 6){\sx{.5}{$2$}}
%\put(70 , 6){\sx{.5}{$3$}}
+
\put(40 , 6){\sx{.5}{$0$}}
%\put(78 , 6){\sx{.5}{$\Re(z)$}}
+
\put(17 ,6){\sx{.5}{$-\!2$}}
\put(70 , 6){\sx{.5}{$x$}}
+
\put(31,56){\sx{.5}{$D\!>\!14$}}
  +
\put(68,20){\sx{.5}{$D\!<\!1$}}
}
 
%\sx{2.33}{\begin{picture}(96,75)
 
\sx{2.33}{\begin{picture}(80,75)
 
%\put(0,0){\includegraphics{figtaifima}}
 
\put(0,0){\includegraphics{vladi07a}}
 
\tafiax
 
\put(31,54){\sx{.45}{$D\!>\!14$}}
 
\end{picture}}
 
\sx{2.33}{\begin{picture}(84,70)
 
%\put(0,0){\includegraphics{figtaimaclo}}
 
\put(0,0){\includegraphics{vladi07b}}
 
\tafiax
 
\put(33,55){\sx{.52}{$D\!<\!1$}}
 
\put(33,24){\sx{.45}{$D\!>\!14$}}
 
 
\end{picture}}
 
\end{picture}}
   
 
\end{document}
 
\end{document}
 
</nowiki></nomathjax></poem>
 
</nowiki></nomathjax></poem>
  +
   
 
[[Category:Agreement]]
 
[[Category:Agreement]]
 
[[Category:Book]]
 
[[Category:Book]]
 
[[Category:Agreement]]
 
[[Category:Agreement]]
[[Category:Complex map]]
 
 
[[Category:BookMap]]
 
[[Category:BookMap]]
 
[[Category:Tetration]]
 
[[Category:Tetration]]

Latest revision as of 08:56, 1 December 2018

Agreement of approximations of tetration by elementary functions with the original representation through the Cauchi integral is shown in the left hand side map.

The similar map at the right hand side represents the comparison to the similar representation through the Cauchi integral, but the contour of integration is displaced for 1/2 to the left.

Levels $D=\mathrm{const}$ are drawn with step 2, but the exception is dome for level $D=1$, this level is drawn with thick lines.

$\displaystyle {\rm fse}(z)=\left\{ \begin{array}{ccrcccr} {\rm fima}(z) &,& 4.5& \!<\! &\Im(z) \\ {\rm tai}(z) &,& 1.5& \!<\! &\Im(z) &\!\le\!& 4.5 \\ {\rm maclo}(z) &,& -1.5& \le &\Im(z) &\!\le\!& 1.5 \\ {\rm tai}(z^{*})^{*} &,& -4.5& \le &\Im(z) &\!<\!&\!\! -1.5 \\ {\rm fima}(z^{*})^{*} &,& & &\Im(z) &\!<\!&\!\! -4.5 \end{array} \right.$

Mnemonics for the name of the approximation: Fast Super Exponent.

The agreement plotted is

$\displaystyle D=D_{8}(z)=-\lg\left( \frac

\right) $ where $F_4$ denotes the 4th ackermann, id set, natural tetration evaluated through the Cauchi integral [1]. Usage: this is figure 14.11 of the book Суперфункции (2014, In Russian) [2]; the English version is in preparation in 2015. First time published in the Vladikavkaz Matehmatical Journal[3].

Refereces

  1. http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
    http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf
    http://mizugadro.mudns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
  2. https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
    http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
    http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.
  3. http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45. Figure 8.

C++ generator of the first picture

Fsexp.cin, ado.cin, conto.cin, plodi.cin 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++)
#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 "superex.cin"
#include "fsexp.cin"
#include "f4natu.cin"
//z_type FSEXP(z_type z){DB y=Im(z);

z_type fsexp(z_type z){DB y=Im(z);
 if(y> 4.5) return fima(z);
 if(y> 1.5) return tai3(z);
 if(y>-1.5) return maclo(z);
 if(y>-4.5) return conj(tai3(conj(z)));
            return conj(fima(conj(z)));
}
//#include "superlo.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
z_type Zo=z_type(.31813150520476413, 1.3372357014306895);
z_type Zc=z_type(.31813150520476413,-1.3372357014306895);

  int M=100,M1=M+1;
  int N=101,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("figcf4small.eps","w");ado(o,0,0,82,82);
FILE *o;o=fopen("vladi08a.eps","w");ado(o,82,82);
fprintf(o,"41 11 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-4.+.08*(m-.5);
DO(n,N1) Y[n]=-1.+.08*(n-.5);

for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
for(n=0;n<7;n++) { M( -3,n)L(3,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]; printf("run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
        c=fsexp(z);
        d=F4natu(z);
        p=abs(c-d);///(abs(c)+abs(d));
        p=-log(p)/log(10.);
// p=Re(c); q=Im(c);
        if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
        }}
#include"plodi.cin"
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
// system( "gv fig08a.eps");
        system("epstopdf vladi08a.eps");
        system( "open vladi08a.pdf"); // for linux
// getchar(); system("killall Preview");// for macintosh
}

C++ generator of the second picture

vladif5c.cin also should be loaded



#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
#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 "superex.cin"
#include "fsexp.cin"
//#include "f4natu.cin"
#include "vladif5c.cin"
//z_type FSEXP(z_type z){DB y=Im(z);

z_type fsexp(z_type z){DB y=Im(z);
 if(y> 4.5) return fima(z);
 if(y> 1.5) return tai3(z);
 if(y>-1.5) return maclo(z);
 if(y>-4.5) return conj(tai3(conj(z)));
            return conj(fima(conj(z)));
}
//#include "superlo.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
z_type Zo=z_type(.31813150520476413, 1.3372357014306895);
z_type Zc=z_type(.31813150520476413,-1.3372357014306895);

  int M=100,M1=M+1;
  int N=101,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("figcf5small.eps","w");ado(o,0,0,82,82);
FILE *o;o=fopen("vladi08b.eps","w");ado(o,82,82);
fprintf(o,"41 11 translate\n 10 10 scale\n");

DO(m,M1)X[m]=-4.+.08*(m-.5);
DO(n,N1)Y[n]=-1.+.08*(n-.5);

for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
for(n= 0;n<7;n++) { M( -3,n)L(3,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]; printf("run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
        c=fsexp(z);
        d=F5(z);
        p=abs(c-d);///(abs(c)+abs(d));
        p=-log(p)/log(10.);
// p=Re(c); q=Im(c);
        if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
                        }}

#include"plodi.cin"
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi08b.eps");
        system( "open vladi08b.pdf");//linux
//getchar(); system("killall Preview");
}

Latex combiner


\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{geometry}
\paperwidth 378px
%\paperheight 134px
\paperheight 180px
\topmargin -104pt
\oddsidemargin -94pt
\pagestyle{empty}
\begin{document}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}

\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}

\sx{2.2}{\begin{picture}(90,80)
%\put(0,0){\includegraphics{figsexpF4}}
%\put(0,0){\includegraphics{figcf4small}}
\put(0,0){\includegraphics{vladi08a}}
\put(5,68){\sx{.45}{$\Im(z)$}}
\put(7,59){\sx{.5}{$5$}}
\put(7,49){\sx{.5}{$4$}}
\put(7,39){\sx{.5}{$3$}}
\put(7,29){\sx{.5}{$2$}}
\put(7,19){\sx{.5}{$1$}}
\put(7, 9){\sx{.5}{$0$}}
\put(70 , 6){\sx{.45}{$\Re(z)$}}
\put(60 , 6){\sx{.5}{$2$}}
\put(40 , 6){\sx{.5}{$0$}}
\put(17 ,6){\sx{.5}{$-\!2$}}
\put(25,62){\sx{.5}{$D\!>\!14$}}
\put(55,60){\sx{.5}{$12\!<\!D\!<\!14$}}
\put(68,20){\sx{.5}{$D\!<\!1$}}
\end{picture}}
\sx{2.2}{\begin{picture}(80,80)
%\put(0,0){\includegraphics{figsexpF5}}
%\put(0,0){\includegraphics{figcf5small}}
\put(0,0){\includegraphics{vladi08b}}
\put(5,68){\sx{.45}{$\Im(z)$}}
\put(7,59){\sx{.5}{$5$}}
\put(7,49){\sx{.5}{$4$}}
\put(7,39){\sx{.5}{$3$}}
\put(7,29){\sx{.5}{$2$}}
\put(7,19){\sx{.5}{$1$}}
\put(7, 9){\sx{.5}{$0$}}
\put(70 , 6){\sx{.45}{$\Re(z)$}}
\put(60 , 6){\sx{.5}{$2$}}
\put(40 , 6){\sx{.5}{$0$}}
\put(17 ,6){\sx{.5}{$-\!2$}}
\put(31,56){\sx{.5}{$D\!>\!14$}}
\put(68,20){\sx{.5}{$D\!<\!1$}}
\end{picture}}

\end{document}

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