File:SuTraAsyQplotT.png

Explicit plot of function SuTra and its Abelexponential asymptotic.

Desciption

The thick curves represent the following Special functions.

Blue 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 is described at Mathematics of Computation ^[1], 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).

Pink curve: \(\ y=\mathrm{SuTra}(x)\ \). Function SuTra is Superfunction of the Trappmann function \(\ z\mapsto z+\exp(z)\ \). SuTra is Entire Function with Logarithmic Asymptotic, it is described at Applied Mathematical Sciences ^[2], 2013.

Green curve: \(\ y = \mathrm{AuExp_{\sqrt{2},4,5}}\big(\mathrm{SuTra}(x)\big)\ \). At large values of input, this combined function approaches the linear function with tangent unity, \(y = x\!+\!x_{\mathrm{stq2}} \ \).

Red curve: \( y=10 \ \delta(x)\ \). This \(\delta\) it is just difference between the "green curve" and the linear function mentioned: \(\delta(x) = \mathrm{AuExp_{\sqrt{2},4,5}}\big(\mathrm{SuTra}(x)\big) - (x\!+\!x_{\mathrm{stq2}})\). In the picture, this \(\delta\) is almost coincide with the elementary function \( x \mapsto - \exp\!\big( - \pi\ (x\!-\!2.64)\big) \). This exponential is also scaled with factor 10 and shown with thin black curve. This curve almost coincide with the red curve; the deviation is smaller than the thickness of the red curve. The last digit in the constant 2.64 above is subject for the revision (and perhaps the correction).

The special functions mentioned above are described also in book «Superfunctions»^[3], 2020.

The thin black lines represent the following Elementary functions.

Central part of the picture: addition short grid line \(y=\mathrm{SuTra}(2)=1\!+\!\mathrm e\).

Right hand side of the picture: short straight line \(y=x+x_{\mathrm{stq2}}\). At the right hand side edge, it overlaps with the green line.

Left hand side of the picture: curve \( y=-\ln(-x) \); it is just logarithmic asymptotic of function SuTra at large negative values of the real part of its input.

The right hand side, bottom of the picture: \( y= - 10 \exp\!\big( - \pi\ (x\!-\!2.64)\big) \); is strongly overlaps with red curve \( y= 10\ \delta(x) \) mentioned above.

C++

/* subroutines ado.cin, Tania.cin, LambertW.cin, SuZex.cin, Sqrt2f45e.cin, Sqrt2f45l.cin should be loaded in order to compile the source below.*/

/* subroutines ado.cin, Tania.cin, LambertW.cin, SuZex.cin, Sqrt2f45e.cin, Sqrt2f45l.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"
 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.));}
 #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; DB Shift=1.219;  
//FILE *o;o=fopen("48.eps","w"); ado(o,612,1212); fprintf(o,"202 202 translate\n 100 100 scale 1 setlinejoin 2 setlinecap\n");
FILE *o;o=fopen("SuTraAsyQplot.eps","w"); ado(o,612,1212); 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,331){x=2.+.01*m;  z=sutra(x); 
	if(!(abs(z)<1.e24)) {
		// printf("%lg %lg %lg\n",x,Re(z),Im(z));
		break;
		}
	   z=F45L(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>5.1||y>10) break;
	} fprintf(o,".05 W 0 1 0 RGB S\n");
 
DO(m,84){ x=-2.1+.1*m; z=F45E(x+Shift); y=Re(z); if(m==0) M(x,y) else L(x,y); if(y>10) break; }  fprintf(o,".03 W 0 0 1 RGB S\n"); 

DO(m,403){x=2.+.01*m; z=sutra(x); if(!(abs(z)<1.e19)) 
		{
		//	printf("%lg %lg %lg\n",x,Re(z),Im(z)); 
			break;
		} 
	z=Re(F45L(z)); 	y=Re(z); 
	// printf("%6.3lf %16.14lf \n",x,Re(z));
			y-=x; y-=Shift; // 1.219; 
	y*=10;
	if(m==0) M(x,y) else L(x,y) if(x>6.03||y>6) break;
	} 
fprintf(o,".04 W 1 0 0 RGB S\n"); 

//DO(m,111){x=3.+.02*m; y=-exp(-M_PI*(x-3.37)); if(m==0) M(x,y) else L(x,y) } fprintf(o,".01 W 0 0 0 RGB S\n"); 
DO(m,111){x=3.+.02*m; y=-exp(-M_PI*(x-2.64)); y*=10; if(m==0) M(x,y) else L(x,y) } fprintf(o,".01 W 0 0 0 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(2,2+Shift)L(5,5+Shift)
M(2,3.219)L(5,6.219)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 SuTraAsyQplot.eps"); 
      system(    "open SuTraAsyQplot.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( 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(308,832){\sx{5}{\rot{70}\(y\!=\!\mathrm{Asym}(x)\)\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}\(y=x\!+\!x_{\mathrm{stq2}}\)\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(560,36){\sx{3}{\rot{68}\(y=10 \ \delta(x)\)\ero}}
\end{picture}
\end{document}

References

Keywords

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

Суперфункции]]»,

«ado.cin», «Tania.cin», «LambertW.cin», «SuZex.cin», «Sqrt2f45e.cin», «Sqrt2f45l.cin»