File:SuTraAsyQ2ateT.png

Explicit plot of the generalyzed Asymptotic «Asym» (thick light blue (Cyan) curve) of function SuTra (pink curve) through the growing SuperExponential to base \(\sqrt{2}\) and the related functions.

Asymptotic statement

For real \(x\to +\infty\), \(x>0\), \[ \mathrm{SuTra}(x) \ \underset{\overset{x\to+\infty}{x>0}}{\overset{\mathrm{ate}}{\sim}} \mathrm{Asym}(x) \, , \qquad \mathrm{Asym}(x)= \mathrm{SuExp}_{\sqrt{2},4,5}(x+x_{\mathrm{stq2}}), \] where the shift constant is numerically estimated as \[ x_{\mathrm{stq2}}\approx 1.219 \]

By definition, this means \[ \lim_{x\to+\infty} \Big( \mathrm{ate}\!\big(\mathrm{SuTra}(x)\big) - \mathrm{ate}\!\big(\mathrm{Asym}(x)\big) \Big) =0. \]

Description of the curves

Pink curve: \(\ y=\mathrm{SuTra}(x)\), the superfunction of the Trappmann map \(z\mapsto z+\exp z\). SuTra is Entire Function with Logarithmic Asymptotic, it is described at the Applied Mathematical Sciences ^[1], 2013.

Light blue (cyan) curve: \(\ y=\mathrm{Asym}(x) = \mathrm{SuExp}_{\sqrt{2},4,5}(x\!+\!x_{\mathrm{stq2}}) \) . This is ArctetralAsymptotic of function SuTra with growing SuperExponential to base \(\sqrt{2}\) at large input. This growing SuperExponential \(\mathrm{SuExp}_{\sqrt{2},4,5}\) is described at Mathematics of Computation ^[2], 2010. In this calculus, the following evaluate of the parameter is used: \( x_{\mathrm{stq2}}\!\approx\! 1.219 \ \). In the routine below, identifier "Shift" is used for this parameter. The last digit of this estimate is subject for the revision (and perhaps the correction).

Green curve: \(\ y=\mathrm{ate}\!\big(\mathrm{SuTra}(x)\big)\), the ArcTetrational "image" of SuTra. The natural ArcTetration \(\mathrm{ate}\) is described at Mathematics of Computation ^[3], 2009; the efficient implementations of the natural tetration \(\mathrm{tet}\) and that of the natural ArcTetration \(\mathrm{ate}\) are described at Vladikavkaz Mathematical Journal ^[4], 2010. Even improved algorithms to evaluate the tetration and arctetration are suggested also by William Paulsen and Samuel Patrick Cowgill ^[5][6][7], but, up to year 2026, the C++ implementations for these algorithms are not found in the free access.

Thin black curve at the left hand side: \( y=-\ln(-x)\), it is asymptotic of function SuTra at large negative input.

Dark blue curve: \(\ y=\mathrm{ate}\!\big(\mathrm{Asym}(x)\big)\), the ArcTetrational image of the asymptotic function.

The functions mentioned above (except the «Asym») are described also in book «Superfunctions» ^[8], 2014,2020.

The green and blue curves approach the same straight line, indicating the asymptotic equivalence. This line is approximated with \(\ y=x\!-\!0.7 \ \).

Red curve: \( \ y=\delta(x) \ \); it is arctetral residual \(\ \delta(x)= \mathrm{ate}\!\big(\mathrm{SuTra}(x)\big) - \mathrm{ate}\!\big(\mathrm{Asym}(x)\big)\, \); it tends to zero and quantifies the precision of the asymptotic approximation.

The additional short grid line \(\ y=\mathrm e\!+\!1 = \mathrm{SuTra}(2) \ \) marks the specific value of the SuperTrappmann function at the integer input.

C++

/* subroutines ado.cin, Tania.cin, LambertW.cin, SuZex.cin, fslog.cin should be loaded in order to compile the source 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 "Tania.cin" // need for LambertW
 #include "LambertW.cin" // need for AuZex
 #include "SuZex.cin"
  #include "Sqrt2f45e.cin"
  #include "Sqrt2f45l.cin"
  #include "fslog.cin"
 z_type tra(z_type z){ return exp(z)+z;}
 z_type sutra(z_type z){ if( Re(z)<2. || fabs(Im(z))>2. ) return log(suzex(z));
                                                          return tra(sutra(z-1.));}
DB Shift=1.219; 
z_type Asym(z_type z){ return F45E(z+Shift); }
#include "ado.cin"
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; 
FILE *o;o=fopen("SuTraAsyQ2ate.eps","w"); ado(o,622,1212); fprintf(o,"202 202 translate\n 100 100 scale 1 setlinejoin 2 setlinecap\n");
 DO(m,411){x=-2.02+.02*m; y=Re(sutra(x)); if(m==0) M(x,y) else L(x,y) if(x>6.03||y>10) break;} fprintf(o,".06 W 1 0 1 RGB S\n");
 DO(m,100){x=-2.01+.02*m; y=-log(-x);     if(m==0) M(x,y) else L(x,y) if(x>6.03||y>10) break;} fprintf(o,".01 W 0 0 0 RGB S\n"); 
 DO(m,720){x=-2.1+.01*m;  z=sutra(x); 
	if(!(abs(z)<1.e32)) {
		// printf("%lg %lg %lg\n",x,Re(z),Im(z));
		break;
		}
	   z=FSLOG(z); y=Re(z);	//printf("%6.3lf %16.14lf \n",x,Re(z));
	if(m==0) M(x,y) else L(x,y) 	if(x>4.4||y>10) break;
	} fprintf(o,".06 W 0 1 0 RGB S\n");
 
 DO(m,720){x=-2.1+.01*m;  z=Asym(x); 
	if(!(abs(z)<1.e32)) { 		// printf("%lg %lg %lg\n",x,Re(z),Im(z));
				break;		}
	//   z=F45L(z); y=Re(z);
	   z=FSLOG(z); y=Re(z); //printf("%6.3lf %16.14lf \n",x,Re(z));
	if(m==0) M(x,y) else L(x,y) 	if(x>4.4||y>10) break;
	} fprintf(o,".04 W 0 0 1  RGB S\n");

 DO(m,637){x=-2.1+.01*m; c=FSLOG(sutra(x)); d=FSLOG(Asym(x)); y=Re((c-d)); if(m==0) M(x,y) else L(x,y); } 
fprintf(o,".03 W 1 0 0 RGB S\n");
 

DO(m,84){ x=-2.1+.1*m; z=Asym(x); y=Re(z); if(m==0) M(x,y) else L(x,y); if(y>10) break; }  fprintf(o,".09 W 0 1 1 RGB S\n"); 

 for(n=-2;n<11;n++) {M(-2,n)L(4,n)}
 for(m=-2;m<5;m++) {M(m,-2)L(m,10)}   M(0,1.+M_E) L(2,1.+M_E)  
M(-1,-1.7)L(5,5-.7)fprintf(o,"0 0 0 RGB .004 W S\n");
M(0,-2.1)L(0,10.1)
M(-2.1,0)L(6.1,0) fprintf(o,".03 W S\n");

fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%');  fclose(o);
      system("epstopdf SuTraAsyQ2ate.eps"); 
      system(    "open SuTraAsyQ2ate.pdf"); //for macintosh
return 0;

Latex

\documentclass[12pr]{article}
\paperwidth 646pt
\paperheight 1240pt
\textwidth 800pt
\textheight 1700pt
\topmargin -96pt
\oddsidemargin -66pt
%\usepackage{xcolor}
\usepackage{graphicx}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \ing {\includegraphics}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\parindent 0pt
\begin{document}
\begin{picture}(720,1220)
%\put(-2,-6){\textcolor{white}{\rule{740pt}{740pt}}}
%\put(20,20){\ing{SuTraAsy3plot}}
%\put(20,20){\ing{SuTraAsyQplot}}
\put(20,20){\ing{SuTraAsyQ2ate}}
\put( 3,1206){\sx{3.2}{\(y\)}}
\put( 4,1116){\sx{3}{\(9\)}}
\put( 4,1016){\sx{3}{\(8\)}}
\put( 4,916){\sx{3}{\(7\)}}
\put( 4,816){\sx{3}{\(6\)}}
\put( 4,716){\sx{3}{\(5\)}}
\put( 4,616){\sx{3}{\(4\)}}
\put(240,585){\sx{4}{\(y\!=\!1\!+\!\mathrm e\)}}
\put( 4,516){\sx{3}{\(3\)}}
\put( 4,416){\sx{3}{\(2\)}}
\put( 4,316){\sx{3}{\(1\)}}
\put( 4,216){\sx{3}{\(0\)}}
\put(-6,114){\sx{2.4}{\(-\!1\)}}
\put(-6,14){\sx{2.4}{\(-\!2\)}}
\put(0,-6){\sx{2.8}{\(-2\)}}
\put(102,-5){\sx{2.9}{\(-1\)}}
\put(216,-5){\sx{3}{\(0\)}}
\put(316,-5){\sx{3}{\(1\)}}
\put(416,-5){\sx{3}{\(2\)}}
\put(516,-5){\sx{3}{\(3\)}}
\put(606,-4){\sx{3.6}{\(x\)}}
%\put(190,356){\sx{3}{\rot{83}\(y\!=\!-\ln(-x)\)\ero}}
\put(68,188){\sx{3}{\rot{48}\(y\!=\!-\ln(-x)\)\ero}}
\put(330,886){\sx{4.5}{\rot{73}\(y\!=\!\mathrm{Asym}(x)\)\ero}}
\put(455,980){\sx{4.4}{\rot{86}\(y\!=\!\mathrm{SuTra}(x)\)\ero}}

\put(170,360){\sx{4.4}{\rot{13}\(y\!=\!\mathrm{ate}\!\big(\mathrm{Asym}(x)\big)\)\ero}}

%\put(488,728){\sx{5}{\rot{85}\(y\!=\!\mathrm{SuTra}(x)\)\ero}}
%\put(24,119){\sx{3}{\rot{22}\(y\!=\!\mathrm{SuTra}(x)\)\ero}}
%\put(488,633){\sx{3.4}{\rot{44}\(y=x\!+\!x_{\mathrm{ste2}}\)\ero}}
\put(270,208){\sx{3.4}{\rot{44}\(y=x\!+\!0.7\)\ero}}
\put(298,160){\sx{4}{\rot{47}\(y\!=\!\mathrm{ate}\!\big(\mathrm{SuTra}(x)\big)\)\ero}}
%\put(440,56){\sx{4}{\rot{84}\(y\!=\!\mathrm{AuExp_{\sqrt{2},4,5}}\big(\mathrm{SuTra}(x)\big)\)\ero}}

%\put(528,62){\sx{3}{\rot{70}\(y=10 \ \delta(x)\)\ero}}
\put(396,125){\sx{4.4}{\rot{28}\(y\!=\!\delta(x)\)\ero}}
\end{picture}
\end{document}

References

Keywords

«Abelfunction», «Approximation», «Arctetral asymptoric», «Asymptotic», «Base sqrt2», «Logarithmic asymptotic», «Natural ArcTetration», «SuperExponential», «Superfunction», «Superfunctions», «SuTra», «Trappmann function»,

«Суперфункции»,

«ado.cin», «fsexp.cin» (C++ implementation of natural tetration), «fslog.cin» (C++ implementation of natural ArcTetration), «Tania.cin», «LambertW.cin», «SuZex.cin» (Function SuTra is implemented through function SuZex\( = z\mapsto z\ \exp(z)\)), «Sqrt2f45e.cin» (C++ implementation of growing SuperExponential to base sqrt2), «Sqrt2f45l.cin» (C++ implementation of growing AbelExponential to base sqrt2)