File:Vladi10.jpg

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Complex map of approximation slo of the natural arctetration ate and agreements at the substitution of this approximation into the transfer equation.



Usage: this is figure 15.2 of the book Суперфункции (2014, In Russian) [1]; the English version is in preparation in 2015.

First time published in the Vladikavkaz Matehmatical Journal [2].

Refereces

  1. ↑ 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.
  2. ↑ 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 10.


C++ generator of the compelx map

ado.cin, conto.cin, fslog.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 "fslog.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=200,M1=M+1;
  int N=402,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
//DB X[201],Y[403], g[40803],f[40803], w[40803]; // w is working array.
DB X[201],Y[403], g[81003],f[81003], w[81003]; // w is working array.
char v[81003]; // v is working array
//FILE *o;o=fopen("figslo.eps","w");ado(o,0,0,62,62);
FILE *o;o=fopen("vladi10a.eps","w");ado(o,62,62);
fprintf(o,"21 31 translate\n 10 10 scale\n");

z_type L=z_type(.31813150520476413, 1.3372357014306895);
p=Re(L); q=Im(L); DB R=abs(L); DB A=arg(L);
fprintf(o,"0 0 %9.6f %9.6f %9.6f arc C .5 1 0 RGB F\n",R,-180/M_PI*A,180/M_PI*A);

// DB sx=3./sinh(.01*N);
// DO(m,M1) X[m]=sx*sinh(.02*(m-M/2-.5));
DO(m,M1) X[m]=-1.+.02*(m-.5);
//DO(n,N1)Y[n]=-3.3+.014*(n-.5);
  DO(n,N1)Y[n]=-2.+.01*(n-.5);
//DB sy=6./sinh(.01*N);
//DO(n,N1) Y[n]=sy*sinh(.02*(n-N/2-.5));

for(m=-2;m<4;m++) { if(m==0){M(m,-2.1)L(m,2.1)}
                        else {M(m,-2)L(m,2)} }
for(n=-2;n<3;n++) {M( -2,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; }
//z_type F[M1*N1];
DO(m,M1){x=X[m]; printf("50 run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
         c=slo(z);
        p=Re(c); q=Im(c);
        if(p>-9 && p<9 && q>-9 && q<9 ) {g[m*N1+n]=p;f[m*N1+n]=q;}
        }}
p=.5;q=.4;
//p=2;q=1;
fprintf(o,"1 setlinecap 1 setlinejoin\n");

for(m=-19;m<19;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".006 W 0 .6 0 RGB S\n");
for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);
                                                fprintf(o,".006 W .9 0 0 RGB S\n");
for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".006 W 0 0 .9 RGB S\n");

for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".024 W .8 0 0 RGB S\n");
for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".024 W 0 0 .8 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".024 W .5 0 .5 RGB S\n");
for(m=-31;m<32;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".024 W 0 0 0 RGB S\n");

//#include"plofu.cin"

fprintf(o,"0 setlinecap 0 setlinejoin\n");

M(Re(Zo),Im(Zo)) L(-2.02,Im(Zo))
M(Re(Zo),-Im(Zo)) L(-2.02,-Im(Zo))
fprintf(o,".02 W 1 1 1 RGB 0 setlinecap S\n");
DO(m,8){ M(Re(Zo)-.31*(m ), Im(Zo)) L(Re(Zo)-.31*(m+.5), Im(Zo)) }
DO(m,8){ M(Re(Zo)-.31*(m ),-Im(Zo)) L(Re(Zo)-.31*(m+.5),-Im(Zo)) }
fprintf(o,".05 W 0 0 0 RGB S\n");
// fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n");
//setdash is not supported
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi10a.eps");
        system( "open vladi10a.pdf");
getchar(); system("killall Preview");
}

C++ generator of nap of agreement $D_{\mathrm A}$

ado.cin, conto.cin, fslog.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;
//#include <complex.h>
//#define z_type complex<double>
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)

#include "fslog.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=200,M1=M+1;
  int N=201,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("figsloE.eps","w");ado(o,0,0,52,52);
FILE *o;o=fopen("vladi10b.eps","w");ado(o,52,52);
fprintf(o,"21 31 translate\n 10 10 scale\n");

z_type L=z_type(.31813150520476413, 1.3372357014306895);
p=Re(L); q=Im(L); DB R=abs(L); DB A=arg(L);
fprintf(o,"0 0 %9.6f %9.6f %9.6f arc C .5 1 0 RGB F\n",R,-180/M_PI*A,180/M_PI*A);

DO(m,M1) X[m]=-1.2+.024*(m-.5);
//DO(n,N1) Y[n]=-1.2 +.02*n*(1.+.000008*(n-8.)*(n-8));
DO(n,N1) Y[n]=-2.2 +.024*n;

//DB sy=6./sinh(.005*N);
//DO(n,N1) Y[n]=sy*sinh(.01*(n-N/2-.5+10));

for(m=-2;m<4;m++) { if(m==0){M(m,-2.1)L(m,2.08)}
                        else {M(m,-2)L(m,2)} }
for(n=-2;n<3;n++) {M( -2,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; }
for(m=0;m<M1;m++){x=X[m]; printf("50 run at x=%6.3f\n",x);
                        DO(n,N1){y=Y[n]; z=z_type(x,y);
                                c=slo(exp(z))-1.;
                                d=slo(z);
                                p=abs(c-d);
                                p=-log(p)/log(10.);
                                // p=Re(log(c))/log(10.);
                                if(p>-99 && p<99) g[m*N1+n]=p;
                        }}
#include"plodi.cin"
//M(-10,0)L(-2,0)fprintf(o,".04 W 0 0 0 RGB [.1 .1] 0 setdash S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi10b.eps");
        system( "open vladi10b.pdf");
getchar(); system("killall Preview");
}

C++ generator of nap of agreement $D_{\mathrm B}$

ado.cin, conto.cin, fslog.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 "fslog.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=200,M1=M+1;
  int N=201,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("vladi10c.eps","w");ado(o,52,52);
fprintf(o,"21 31 translate\n 10 10 scale\n");

z_type L=z_type(.31813150520476413, 1.3372357014306895);
p=Re(L); q=Im(L); DB R=abs(L); DB A=arg(L);
fprintf(o,"0 0 %9.6f %9.6f %9.6f arc C .5 1 0 RGB F\n",R,-180/M_PI*A,180/M_PI*A);

DO(m,M1) X[m]=-1.2+.024*(m-.5);
//DO(n,N1) Y[n]=-1.2 +.02*n*(1.+.000008*(n-8.)*(n-8));
DO(n,N1) Y[n]=-2.2 +.024*n;

//DB sy=6./sinh(.005*N);
//DO(n,N1) Y[n]=sy*sinh(.01*(n-N/2-.5+10));

for(m=-2;m<4;m++) { if(m==0){M(m,-2.1)L(m,2.08)}
                        else {M(m,-2)L(m,2)} }
for(n=-2;n<3;n++) {M( -2,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("50 run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
        c=slo(log(z))+1.;
        d=slo(z);
        p=abs(c-d);
        p=-log(p)/log(10.);
        // p=Re(log(c))/log(10.);
        if(p>-99 && p<99) g[m*N1+n]=p;
        }}
#include"plodi.cin"
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi10c.eps");
        system( "open vladi10c.pdf");//linux
//getchar(); system("killall Preview");//macintosh
}

Latex combiner


\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{geometry}
\paperwidth 418px
%\paperheight 134px
\paperheight 105px
\topmargin -106pt
\oddsidemargin -94pt
\pagestyle{empty}
\begin{document}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}

\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \fsloax {
\put(-2.3,49){\sx{.45}{$y$}}
\put(-2.3,39.6){\sx{.45}{$1$}}
\put(-2.3,29.6){\sx{.45}{$0$}}
\put(-6,19.6){\sx{.45}{$-1$}}
\put(-6, 9.6){\sx{.45}{$-2$}}
\put( 7 ,6.6){\sx{.45}{$-\!1$}}
\put(19.8 , 6.6){\sx{.45}{$0$}}
\put(30 , 6.6){\sx{.45}{$1$}}
\put(40 , 6.6){\sx{.45}{$2$}}
\put(48.6 , 6.6){\sx{.45}{$x$}}
}
\hskip 17pt
%\sx{2.75}{\begin{picture}(60,55)
\sx{2.34}{\begin{picture}(60,50)
\put(0,0){\includegraphics{vladi10a}}
\fsloax
\put(32.2,25.8){\rot{90}\sx{.4}{$u\!=\!0$}\ero}
\put(37,29.8){\sx{.4}{$v\!=\!0$}}
\end{picture}}
\sx{2.34}{\begin{picture}(60,50)
%\put(0,0){\includegraphics{figsloE}}
\put(0,0){\includegraphics{vladi10b}}
\fsloax
\put(32,44){\sx{.5}{$D_{\rm A}\!<\!1$}}
\put(21,33){\sx{.55}{$15$}}
\end{picture}}
\sx{2.34}{\begin{picture}(50,50)
%\put(0,0){\includegraphics{figsloL}}
\put(0,0){\includegraphics{vladi10c}}
\fsloax
\put(3,33){\sx{.5}{$D_{\rm B}\!<\!1$}}
\put(32,33){\sx{.55}{$15$}}
\end{picture}}

\end{document}

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