Difference between revisions of "File:Autran0m9tes64t.jpg"

Fig.20.10 from page 285 of book «Superfunctions»^[1], 2020.

This image is used also as Рис.20.10 at page 295 of the Russian version «Суперфункции» ^[2], 2014.

The figure shows the precision of the primary approximation through the asymptotic for function AuTra. It is Abelfunction for the elementary Trappmann function \(z \mapsto z+\exp(z) \).

The contours show the Agreement \( \mathcal A=\mathcal A(x+\mathrm i y)\) for the primary approximation \(g\) of function AuTra with \(M=9\): \[ \mathcal A(z)= -\lg\left( \frac {|\mathrm{SuTra}(g_M(z))-z|} {|\mathrm{SuTra}(g_M(z))|+|z|} \right) \] The primary approximation \(g_M\) is expressed through the asymptotic \[ G_M(z)=\frac{z}{2}-\mathrm e^{-z} + \sum_{m=1}^{M}c_m \mathrm e^{mz} \] The coefficients \(c_m\) can be evaluated from the Abel equation, "minimizing" the residual at the substutution of G_M into equation \[ G(\mathrm{tra}(z))=G(z)+1 \] The AbelTrappmann function \[ G(z)=\lim_{n\to \infty} G_M(\mathrm{ArcTra}^n(z))+n) \] Then \[ \mathrm{AuTra}(z)=G(z)-G(0) \approx G(z) + 1.1259817765745026 \] The primary approximation \[ g_M(z)=G_M(z)+1.1259817765745026 \] The agreement \(\mathcal A \) shows the range where the primary approzimtion can be used "as is" for evaluation of function SuTra.

The chosen region is shaded with light green.

The shaded region shows the range \[ |y|<3 \ ,\ |y|/3+x<3.2 \]

For other values, for the implementation, the Abel equation is applied recursively, in order to express the value of SuTra in terms of the value from the shaded region.

Functions AuTra and SuTra are described also at the «Applied Mathematical Sciences» ^[3], 2013.

C++ generator of curves

/* Files autran.cin,sutran.cin, ado.cin, conto.cin shold 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 complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
z_type tra(z_type z) {return z+exp(z);}  

//#include "tania.cin"
//#include "LambertW.cin"
//#include "auzex.cin"

z_type arctra(z_type z){return z-Tania(z-1.);} 
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.; }

#include "sutran.cin"
#include "arctran.cin"

z_type autran0(z_type z) {z_type e=exp(z); z_type s; int n,M;
DB c[20]={-0.166666666666666667, 0.062500000000000, -0.0351851851851851852,  0.0208333333333333333,
-0.00976190476190476190, 0.000356867283950617284, 0.00577884857646762409, -0.0054935515873015873,
-0.00258505283582444076, 0.0121986400462962963, -0.00649411105018518438, -0.0264514796679871911,
 0.0478515524404502325, 0.0537587298747943833, -0.270736261932081259, -0.00655211866410402040,
 1.62788126261366988, -1.60768769009409886, -10.8381871746651334, 24.7850929105834429};
//M=14;
M=9;
s=c[M];
for(n=M-1;n>=0;n--) { s*=e; s+=c[n];}
                         return z/2.-1./e +e*s + 1.1259817765745026;}

z_type autran(z_type z) 
{ int n; z_type s=z; DB y; 
   DO(n,400){ y=fabs(Im(s)); 
              if( y<3. && y < 1.5*(-2.5-Re(s))  ) return autran0(s)+(0.+n);
              s=arctran(s);
            }
   printf("autran: z=%9.6f %9.7f s=%9.6f %9.6f \n",Re(z),Im(z),Re(s),Im(s) );
 getchar(); return 0.;
}
 
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=200,M1=M+1;
int N=1600,N1=N+1;
DB X[M1],Y[N1]; DB g[M1*N1]; DB f[M1*N1]; DB w[M1*N1]; char v[M1*N1];

FILE *o;o=fopen("autran0m9tes64.eps","w");  ado(o,602,802);
fprintf(o,"501 401 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-5.+.028*(m-.5);
DO(n,N1) Y[n]=-4.+.005*(n-.5); 

M(-5.1,3) L(-3.5,3) L(-2.5,0) L(-3.5,-3) L(-5.1,-3)
fprintf(o,"C .7 1 .7 RGB F\n");

for(m=-5;m<2;m++){M(m,-4) L(m,4)  }
for(n=-4;n<5;n++){M(  -5,n) L(1,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=autran0(z);
  c=sutran(c);
 p=abs(c-z)/(abs(c)+abs(z)); p=-log(p)/log(10.); 
 if(p>.5 && p<20.){ 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=.1;
 conto(o,g,w,v,X,Y,M,N,(16.  ),-p,p); fprintf(o,".003 W 1 0 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(15.  ),-p,p); fprintf(o,".012 W .6 0 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14.  ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(13.  ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(12.  ),-p,p); fprintf(o,".012 W 0 0 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(11.  ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(10.  ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (9.  ),-p,p); fprintf(o,".014 W 0 .5 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (8.  ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (7.  ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (6.  ),-p,p); fprintf(o,".012 W 0 .6 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (5.  ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (4.  ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (3.  ),-p,p); fprintf(o,".012 W 1 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (2.  ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (1.  ),-p,p); fprintf(o,".015 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 autran0m9tes64.eps"); 
      system(    "open autran0m9tes64.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh

c=autra( z_type(-1.,M_PI) );
z=sutran(c);
printf("c=%19.16lf %19.16lf z=%19.16lf %19.16lf\n",Re(c),Im(c),Re(z),Im(z)); 
}

Latex generator of labels

\documentclass[12pt]{article} 
\paperwidth 602px 
\paperheight 802px 
\textwidth 3004px 
\textheight 3000px 
\topmargin -106px 
\oddsidemargin -78px 
\usepackage{graphics} 
\usepackage{rotating} 
\newcommand \sx {\scalebox} 
\newcommand \rot {\begin{rotate}} 
\newcommand \ero {\end{rotate}} 
\newcommand \ing {\includegraphics} 
\newcommand \rmi {\mathrm{i}} 
\parindent 0pt
\pagestyle{empty}
\begin{document} 
\parindent 0pt 
\hskip 6pt
\begin{picture}(610,802)
%\put(1,1){\ing{logq2map}}
\put(0,0){\ing{autran0m9tes64}}
\put(18,780){\sx{4}{$y$}}
\put(18,688){\sx{4}{$3$}}
\put(18,588){\sx{4}{$2$}}
\put(18,488){\sx{4}{$1$}}
\put(18,388){\sx{4}{$0$}}
\put(-6,288){\sx{4}{$-1$}}
\put(-6,188){\sx{4}{$-2$}}
\put(-6,088){\sx{4}{$-3$}}
%\put(590,2){\sx{.7}{\rot{80}$v\!=\!4$\ero}}
\put(078,3){\sx{3}{$-4$}}
\put(178,3){\sx{3}{$-3$}}
\put(278,3){\sx{3}{$-2$}}
\put(378,3){\sx{3}{$-1$}}
\put(494,3){\sx{3}{$0$}}
\put(580,3){\sx{3.6}{$x$}}

\put(210,750){\sx{2.8}{$\mathcal A \!<\! 1$}}
\put(276,350){\sx{3}{\rot{90}$\mathcal A\!=\!15$\ero}}
\put(332,350){\sx{3}{\rot{90}$\mathcal A\!=\!12$\ero}}
\put(386,360){\sx{3}{\rot{90}$\mathcal A\!=\!9$\ero}}
\put(440,360){\sx{3}{\rot{90}$\mathcal A\!=\!6$\ero}}
\put(500,260){\sx{3}{\rot{90}$\mathcal A\!=\!3$\ero}}
\put(542,260){\sx{3}{\rot{90}$\mathcal A\!=\!1$\ero}}
\put(528,390){\sx{2.8}{$\mathcal A \!<\! 1$}}

%\put(111,412){\sx{3}{$\mathcal A > 16$}}
\put(166,142){\sx{2.8}{$\mathcal A \!\approx\! 16$}}
%\put(530,155){\sx{2.8}{$\mathcal A \!<\! 1$}}
\end{picture}
\end{document}

References

Keywords

«Abelfunction», «Agreement», «AuTra», «Entire Function with Logarithmic Asymptotic», «Iterate», «Primary apprioximation», «SuTra», «Superfunctions», «Trappmann function»,