Difference between revisions of "File:Expe1eplotT.jpg"

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[[Explicit plot]] of [[exponential]] to [[base e1e]] (thick green curve) and
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Importing image file
  
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that of the [[exponential]] to [[base sqrt2]] (thin red curve)
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Here, $\eta\!=\!\exp(1\mathrm e)\!\approx1.44466786$\! is the Henryk base. At this base the exponential has only one real [[fixed point]],
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id est, equation $\exp_\eta(L)\!=\!L$ has only one real solution $L\!=\!\mathrm e\!\approx\! 2.71~$.
  +
  +
The thick green curve is $!y\!=\!\eta^x$.
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  +
In order to show the fixed point, the thin line $y\!=\!x$ is drawn.
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  +
For comparison, the exponential to base $b\!=\! \sqrt{2}$ is plotted, that has two fixed points, $L\!=\!2$ and $L\!=\!4$.
  +
  +
==[[C++]] generator of curves==
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//<poem><nomathjax><nowiki>
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#include<math.h>
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#include<stdio.h>
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#include<stdlib.h>
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#define DB double
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#define DO(x,y) for(x=0;x<y;x++)
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#include "ado.cin"
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DB B=sqrt(2.);
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int main(){ int m,n; double x,y; FILE *o;
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o=fopen("expe1eplot.eps","w"); ado(o,1204,804);
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fprintf(o,"602 2 translate 100 100 scale\n");
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#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
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#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
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for(m=-6;m<7;m++) {M(m,0)L(m,8)}
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for(m=0;m<9;m++) {M(-6,m)L(6,m)}
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fprintf(o,"2 setlinecap .01 W S\n 1 setlinejoin \n");
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M(M_E,0)L(M_E,M_E)L(0,M_E) fprintf(o,".007 W S\n");
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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,".02 W .8 0 0 RGB S\n");
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for(m=0;m<123;m++){x=-6.1+.1*m; y=exp(x/M_E); if(m==0)M(x,y) else L(x,y);} fprintf(o,".04 W 0 .6 0 RGB S\n");
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M(-.1,-.1)L(6.1,6.1) fprintf(o,".016 W 0 0 0 RGB S\n\n");
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fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
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system("epstopdf expe1eplot.eps");
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system( "open expe1eplot.pdf");
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getchar(); system("killall Preview");//for mac
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}
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//</nowiki></nomathjax></poem>
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==[[Latex]] generator of labels==
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%<poem><nomathjax><nowiki>
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\documentclass[12pt]{article}
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\usepackage{geometry}
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\usepackage{graphicx}
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\usepackage{rotating}
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\paperwidth 1212pt
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\paperheight 844pt
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\topmargin -92pt
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\oddsidemargin -80pt
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\textwidth 1604pt
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\textheight 1604pt
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\pagestyle {empty}
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\newcommand \sx {\scalebox}
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\newcommand \rot {\begin{rotate}}
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\newcommand \ero {\end{rotate}}
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\newcommand \ing {\includegraphics}
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\parindent 0pt
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\pagestyle{empty}
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\begin{document}
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{\begin{picture}(1202,802)
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\put(590,792){\sx{4.2}{$y$}}
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\put(590,698){\sx{4.2}{$7$}}
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\put(590,598){\sx{4.2}{$6$}}
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\put(590,498){\sx{4.2}{$5$}}
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\put(590,398){\sx{4.2}{$4$}}
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\put(590,298){\sx{4.2}{$3$}}
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\put(620,274){\sx{4.2}{$\mathrm e$}}
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\put(590,198){\sx{4.2}{$2$}}
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\put(590,098){\sx{4.2}{$1$}}
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\put(080,-22){\sx{4}{$-5$}}
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\put(180,-22){\sx{4}{$-4$}}
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\put(281,-22){\sx{4}{$-3$}}
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\put(381,-22){\sx{4}{$-2$}}
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\put(482,-22){\sx{4}{$-\!1$}}
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\put(603.6,-22){\sx{4}{$0$}}
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\put(703.7,-22){\sx{4}{$1$}}
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\put(803.8,-22){\sx{4}{$2$}}
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\put(877.,16){\sx{4}{$\mathrm e$}}
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\put(903.9,-22){\sx{4}{$3$}}
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\put(1004.0,-22){\sx{4}{$4$}}
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\put(1104.1,-22){\sx{4}{$5$}}
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\put(1192.2,-22){\sx{4.3}{$x$}}
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%\put(0815,520){\sx{5.6}{\rot{78}$y\!=\!\exp(x)$\ero}}
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\put(1118,678){\sx{4.5}{\rot{69}$y\!=\!\eta^x$\ero}}
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%\put(1076,606){\sx{4.1}{\rot{67}$y\!=\!\exp_{\eta}(x)$\ero}}
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%\put(1100,520){\sx{4}{\rot{62}$y\!=\!\exp_{_{\!\!\sqrt{2}}}(x)$\ero}}
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\put(1130,550){\sx{4}{\rot{61}$y\!=\!(\sqrt{2})^x$\ero}}
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\put(1134,488){\sx{5}{\rot{45.1}$y\!=\!x$\ero}}
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\put(10,10){\ing{expe1eplot}}
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\end{picture}}
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\end{document}
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</nowiki></nomathjax></poem>
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[[Category:Book]]
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[[Category:BookPlot]]
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[[Category:Base e1e]]
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[[Category:Base sqrt2]]
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[[Category:Exponential]]
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[[Category:Fixed point]]
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[[Category:Transfer function]]
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[[Category:SuperFunctions]]
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[[Category:AbelFunctions]]
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[[Category:Explicit plot]]
  +
[[Category:C++]]
  +
[[Category:Latex]]

Latest revision as of 08:35, 1 December 2018

Explicit plot of exponential to base e1e (thick green curve) and that of the exponential to base sqrt2 (thin red curve)

Here, $\eta\!=\!\exp(1\mathrm e)\!\approx1.44466786$\! is the Henryk base. At this base the exponential has only one real fixed point, id est, equation $\exp_\eta(L)\!=\!L$ has only one real solution $L\!=\!\mathrm e\!\approx\! 2.71~$.

The thick green curve is $!y\!=\!\eta^x$.

In order to show the fixed point, the thin line $y\!=\!x$ is drawn.

For comparison, the exponential to base $b\!=\! \sqrt{2}$ is plotted, that has two fixed points, $L\!=\!2$ and $L\!=\!4$.

C++ generator of curves

//


#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.);

int main(){ int m,n; double x,y; FILE *o;
 o=fopen("expe1eplot.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 1 setlinejoin \n");
 M(M_E,0)L(M_E,M_E)L(0,M_E) fprintf(o,".007 W S\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,".02 W .8 0 0 RGB S\n");

  for(m=0;m<123;m++){x=-6.1+.1*m; y=exp(x/M_E); if(m==0)M(x,y) else L(x,y);} fprintf(o,".04 W 0 .6 0 RGB S\n");

 M(-.1,-.1)L(6.1,6.1) fprintf(o,".016 W 0 0 0 RGB S\n\n");
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
   system("epstopdf expe1eplot.eps");
   system( "open expe1eplot.pdf");
   getchar(); system("killall Preview");//for mac
 }
//

Latex generator of labels

%


\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphicx}
\usepackage{rotating}
\paperwidth 1212pt
\paperheight 844pt
\topmargin -92pt
\oddsidemargin -80pt
\textwidth 1604pt
\textheight 1604pt
\pagestyle {empty}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\parindent 0pt
\pagestyle{empty}
\begin{document}
{\begin{picture}(1202,802)
\put(590,792){\sx{4.2}{$y$}}
\put(590,698){\sx{4.2}{$7$}}
\put(590,598){\sx{4.2}{$6$}}
\put(590,498){\sx{4.2}{$5$}}
\put(590,398){\sx{4.2}{$4$}}
\put(590,298){\sx{4.2}{$3$}}
\put(620,274){\sx{4.2}{$\mathrm e$}}
\put(590,198){\sx{4.2}{$2$}}
\put(590,098){\sx{4.2}{$1$}}
\put(080,-22){\sx{4}{$-5$}}
\put(180,-22){\sx{4}{$-4$}}
\put(281,-22){\sx{4}{$-3$}}
\put(381,-22){\sx{4}{$-2$}}
\put(482,-22){\sx{4}{$-\!1$}}
\put(603.6,-22){\sx{4}{$0$}}
\put(703.7,-22){\sx{4}{$1$}}
\put(803.8,-22){\sx{4}{$2$}}
\put(877.,16){\sx{4}{$\mathrm e$}}
\put(903.9,-22){\sx{4}{$3$}}
\put(1004.0,-22){\sx{4}{$4$}}
\put(1104.1,-22){\sx{4}{$5$}}
\put(1192.2,-22){\sx{4.3}{$x$}}
%\put(0815,520){\sx{5.6}{\rot{78}$y\!=\!\exp(x)$\ero}}
\put(1118,678){\sx{4.5}{\rot{69}$y\!=\!\eta^x$\ero}}
%\put(1076,606){\sx{4.1}{\rot{67}$y\!=\!\exp_{\eta}(x)$\ero}}
%\put(1100,520){\sx{4}{\rot{62}$y\!=\!\exp_{_{\!\!\sqrt{2}}}(x)$\ero}}
\put(1130,550){\sx{4}{\rot{61}$y\!=\!(\sqrt{2})^x$\ero}}
\put(1134,488){\sx{5}{\rot{45.1}$y\!=\!x$\ero}}
\put(10,10){\ing{expe1eplot}}
\end{picture}}
\end{document}

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