File:BookChap14pic3.png

Figure 14.3 from page 178 of book «Superfunctions»^[1], 2020.

The same picture appears as Рис.14.3 at page 176 of the Russian version «Суперфункции» ^[2], 2014.

The picture shows the

Explicit plot of natural Exponential (blue curve)

and that of Exponential to base \(\sqrt{2}\) (red curve)

The thin straight line is bisector of the first quadrant, it corresponds to the identity function.
The line is drawn to show the real fixed points (2 and 4) of the exponential to base \(\sqrt{2}\) and absence of real fixed points of the natural exponential.

C++ generator of curves

/* File ado.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 "ado.cin"

DB B=sqrt(2.);

main(){ int m,n; double x,y; FILE *o;
o=fopen("ExpQ2ePlot.eps","w"); ado(o,1204,804);
fprintf(o,"602 2 translate 100 100 scale\n");
#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
for(m=-6;m<7;m++) {M(m,0)L(m,8)}
for(m=0;m<9;m++) {M(-6,m)L(6,m)}
fprintf(o,"2 setlinecap .01 W S\n 2 setlinecap 1 setlinejoin \n");
for(m=0;m<123;m++){x=-6.1+.1*m; y=exp(log(B)*x); if(m==0)M(x,y) else L(x,y);} fprintf(o,".04 W .8 0 0 RGB S\n");
M(-.1,-.1)L(6.1,6.1) fprintf(o,".016 W 0 0 0 RGB S\n\n");

for(m=0;m<123;m++){x=-6.1+.07*m; y=exp(x); if(m==0)M(x,y) else L(x,y);} fprintf(o,".05 W 0 0 .9 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
   system("epstopdf ExpQ2ePlot.eps");
   system(    "open ExpQ2ePlot.pdf");            
//   getchar(); system("killall Preview");//for mac
}

References

Keywords

«BaseSqrt2», «Explicit plot», «Exponential», «Fixed point», «Superfunctions»,