File:E1e14z600.jpg

Iterates of exponent to various bases $b$.

$y=\exp_b^n(x)$

for various $n$ at

$b=\mathrm e$, top plot,

$b=\eta=\exp(1/\mathrm r)$, intermediate pllot,

$b=\sqrt{2}$, bottom plot.

C++ generator of map for $b\!=\!\mathrm e$
//#define z_type complex typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)

//b=10 //#include "f4ten.cin"

//b=r=2.71
 * 1) include "fsexp.cin"
 * 2) include "fslog.cin"

//b=2 //#include "f2.cin"

//b=3/2=1.5 //#include "F15.cin"

//b=exp(1/e)=1.44... //#include "e1etf.cin"

//b=sqrt(2)=1.41... //#include "f21E.cin"

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
 * 1) include "conto.cin"

FILE *o;o=fopen("e1e14az.eps","w");ado(o,1420,820); fprintf(o,"410 210 translate\n 100 100 scale\n");

for(m=-4;m<11;m++){M(m,-2)L(m,6)} for(n=-2;n<7;n++){M(-4,n)L(10,n)} fprintf(o,"0 0 0 RGB .004 W S\n");

fprintf(o,"1 setlinecap 1 setlinejoin\n");

k=0;for(m=-40;m<101;m+=1){x=.1*m;z=x;c=exp(exp(z));y=Re(c); if(k==0)M(x,y)else L(x,y);k++; if(y>6)break;} fprintf(o,"0 0 0 RGB .01 W S\n"); //BLACK

k=0;for(m=-40;m<101;m+=1){ x=.1*m;y=exp(x); if(k==0)M(x,y) else L(x,y); k++; if(y>6)break;} fprintf(o,"0 0 0 RGB .01 W S\n"); //BLACK

k=0;for(m=-40;m<101;m+=2){x=.1*m;z=x;c=FSEXP(.9+FSLOG(z)); y=Re(c); if(k==0)M(x,y) else L(x,y); k++;if(y>6)break;} fprintf(o,"0 0 .8 RGB .02 W S\n"); //BLUE

k=0;for(m=0;m<141;m+=2){x=-4.+.098*m;z=x;c=FSEXP(.5+FSLOG(z)); y=Re(c); if(k==0)M(x,y) else L(x,y); k++;if(y>6)break;} fprintf(o,"0 .7 0 RGB .02 W S\n"); // green

k=0;for(m=-40;m<101;m+=2) {x=.1*m;z=x;c=FSEXP(.1+FSLOG(z)); y=Re(c); if(k==0)M(x,y) else L(x,y); k++;if(y>6)break;} fprintf(o,".8 0 0 RGB .02 W S\n"); //RED

M(-2,-2)L(6.1,6.1) fprintf(o,"0 0 0 RGB .01 W S\n"); // BLACK

DO(m,82){x=-2.1+.1*m;z=x;c=FSEXP(.1+FSLOG(z));y=Re(c);if(m==0)M(y,x) else L(y,x);} fprintf(o,".8 0 0 RGB .02 W S\n"); //RED

DO(m,64){x=-2.1+.1*m;z=x;c=FSEXP(.5+FSLOG(z));y=Re(c);if(m==0)M(y,x) else L(y,x);} fprintf(o,"0 .7 0 RGB .02 W S\n"); // green

DO(m,48){x=-2.1+.1*m;z=x;c=FSEXP(.9+FSLOG(z));y=Re(c);if(m==0)M(y,x) else L(y,x);} fprintf(o,"0 0 .8 RGB .02 W S\n"); //BLUE

DO(m,48){x=-2.1 +.094*m;z=x;c=exp(z);y=Re(c); if(m==0)M(y,x)else L(y,x);k++;} fprintf(o,"0 0 0 RGB .01 W S\n"); //black

DO(m,33){x=-2.1+.092*m;z=x;c=exp(exp(z));y=Re(c); if(m==0)M(y,x)else L(y,x);} fprintf(o,"0 0 0 RGB .01 W S\n"); //black

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf e1e14az.eps"); system( "open e1e14az.pdf");           //for macintosh //     getchar; system("killall Preview");   //for macintosh }

C++ generator of map for $b\!=\!\eta=\exp(1/\mathrm e)$
//#define z_type complex typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)

//b=10 //#include "f4ten.cin"

//b=r=2.71 //#include "fsexp.cin" //#include "fslog.cin"

//b=2 //#include "f2.cin"

//b=3/2=1.5 //#include "F15.cin"

//b=exp(1/e)=1.44...
 * 1) include "e1etf.cin"
 * 2) include "e1eti.cin"
 * 3) include "e1egf.cin"
 * 4) include "e1egi.cin"

//b=sqrt(2)=1.41... //#include "f21E.cin"

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
 * 1) include "conto.cin"

FILE *o;o=fopen("e1e14bz.eps","w");ado(o,1420,820); fprintf(o,"410 210 translate\n 100 100 scale\n");

for(m=-4;m<11;m++){M(m,-2)L(m,6)} for(n=-2;n<7;n++){M(-4,n)L(10,n)} M(M_E,0)L(M_E,M_E)L(0,M_E) fprintf(o,"0 0 0 RGB .004 W S\n");

fprintf(o,"1 setlinecap 1 setlinejoin\n");

DO(m,84){x=-4.+.1004*m;z=x;c=exp(exp(z/M_E)/M_E);y=Re(c); if(m==0)M(x,y)else L(x,y);} fprintf(o,"0 0 0 RGB .01 W S\n"); //BLACK

DO(m,90){ x=-4.+.1003*m;y=exp(x/M_E); if(m==0)M(x,y) else L(x,y);} fprintf(o,"0 0 0 RGB .01 W S\n"); //BLACK

DO(m,74){ x=M_E+.1*(m+.5); z=x; c=E1EGF(.9+E1EGI(z));y=Re(c);if(m==0)M(x,y) else L(x,y); if(y>6)break;} DO(m,66){ x=M_E-.103*(m+.5); z=x; c=E1ETF(.9+E1ETI(z));y=Re(c);if(m/2*2==m)M(x,y) else L(x,y);} fprintf(o,"0 0 .8 RGB .02 W S\n"); //BLUE

DO(m,74){ x=M_E+.1*(m+.5); z=x; c=E1EGF(.5+E1EGI(z)); y=Re(c); if(m==0)M(x,y) else L(x,y); if(y>6)break;} DO(m,66){ x=M_E-.103*(m+.5); z=x; c=E1ETF(.5+E1ETI(z));y=Re(c);if(m/2*2==m)M(x,y) else L(x,y); if(y<-2)break;} fprintf(o,"0 .7 0 RGB .02 W S\n"); //GREEN

DO(m,74){ x=M_E+.1*(m+.5); z=x; c=E1EGF(.1+E1EGI(z)); y=Re(c); if(m==0)M(x,y) else L(x,y); if(y>6)break;} DO(m,66){ x=M_E-.1*(m+.5); z=x; c=E1ETF(.1+E1ETI(z));y=Re(c);if(m/2*2==m)M(x,y) else L(x,y); if(y<-2)break;} fprintf(o,".8 0 0 RGB .02 W S\n"); //RED

M(-2,-2)L(6,6) fprintf(o,"0 0 0 RGB .01 W S\n"); // BLACK

DO(m,74){ x=M_E+.1*(m+.5); z=x; c=E1EGF(-.1+E1EGI(z)); y=Re(c); if(m==0)M(x,y) else L(x,y); if(y>6)break;} DO(m,42){ x=M_E-.103*(m+.5); z=x; c=E1ETF(-.1+E1ETI(z));y=Re(c);if(m/2*2==m)M(x,y) else L(x,y);} fprintf(o,".8 0 0 RGB .02 W S\n"); //RED

DO(m,74){ x=M_E+.1*(m+.5); z=x; c=E1EGF(-.5+E1EGI(z)); y=Re(c); if(m==0)M(x,y) else L(x,y); if(y>6)break;} DO(m,66){ x=M_E-.066*(m+.5); z=x; c=E1ETF(-.5+E1ETI(z));y=Re(c);if(m/2*2==m)M(x,y) else L(x,y); if(y<-2)break;} fprintf(o,"0 .7 0 RGB .02 W S\n"); //GREEN

DO(m,74){ x=M_E+.1*(m+.5); z=x; c=E1EGF(-.9+E1EGI(z)); y=Re(c); if(m==0)M(x,y) else L(x,y); if(y>6)break;} DO(m,52){ x=M_E-.046*(m+.5); z=x; c=E1ETF(-.9+E1ETI(z));y=Re(c);if(m/2*2==m)M(x,y) else L(x,y);} fprintf(o,"0 0 .8 RGB .02 W S\n"); //BLUE

DO(m,89){ x=.4+.1*(m+.5); z=x; c=log(z)*M_E; y=Re(c); if(m==0)M(x,y) else L(x,y);} DO(m,89){ x=1.14+.1*(m+.5); z=x; c=log(log(z)*M_E)*M_E; y=Re(c); if(m==0)M(x,y) else L(x,y);} fprintf(o,"0 0 0 RGB .02 W S\n"); //Black

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf e1e14bz.eps"); system( "open e1e14bz.pdf");           //for macintosh getchar; system("killall Preview");  //for macintosh }

C++ generator of map for $b\!=\!\sqrt{2}$
//#define z_type complex typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)

//b=10 //#include "f4ten.cin"

//b=r=2.71 //#include "fsexp.cin" //#include "fslog.cin"

//b=2 //#include "f2.cin"

//b=3/2=1.5 //#include "F15.cin"

//b=exp(1/e)=1.44... //#include "e1etf.cin" //#include "e1eti.cin" //#include "e1egf.cin" //#include "e1egi.cin"

//b=sqrt(2)=1.41... //#include "f23E.cin" //#include "f23L.cin" //#include "f43E.cin" //#include "f43L.cin"
 * 1) include "sqrt2f21e.cin"
 * 2) include "sqrt2f21l.cin"
 * 1) include "sqrt2f45e.cin"
 * 2) include "sqrt2f45l.cin"

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
 * 1) include "conto.cin"

FILE *o;o=fopen("e1e14cz.eps","w");ado(o,1420,820); fprintf(o,"410 210 translate\n 100 100 scale\n");

for(m=-4;m<11;m++){M(m,-2)L(m,6)} for(n=-2;n<7;n++){M(-4,n)L(10,n)} //M(M_E,0)L(M_E,M_E)L(0,M_E) fprintf(o,"0 0 0 RGB .004 W S\n");

fprintf(o,"1 setlinecap 1 setlinejoin\n"); q=log(2.)/2.;

M(-2.1,-2.1)L(6.1,6.1) fprintf(o,"0 0 0 RGB .003 W S\n"); //BLACK

DO(m, 89){x=-4.+.0995*m;z=x;c=exp(exp(z*q)*q);y=Re(c); if(m==0)M(x,y)else L(x,y)} DO(m,93){x=-4+.1*m;y=exp(x*q); if(m==0)M(x,y) else L(x,y);} DO(m, 76){x=-2.+.0995*m;z=x;c=exp(exp(z*q)*q);y=Re(c); if(m==0)M(y,x)else L(y,x)} DO(m, 82){x=-2+.0994*m;y=exp(x*q); if(m==0)M(y,x) else L(y,x);} fprintf(o,"0 0 0 RGB .005 W S\n"); //BLACK

DO(m,66){ x=4. - .123*(m+.5); z=x; c=F21E(.9+F21L(z));y=Re(c);if(m/2*2==m)M(x,y) else L(x,y);} fprintf(o,"0 0 .8 RGB .03 W S\n"); //BLUE

DO(m,33){ x=2. + .1*(m+.5); z=x; c=F45E(.9+F45L(z));y=Re(c);if(m==0)M(x,y) else L(x,y); } fprintf(o,"0 0 .4 RGB .01 W S\n"); //BLUE //ok

DO(m,66){ x=4. - .123*(m+.5); z=x; c=F21E(.5+F21L(z));y=Re(c);if(m/2*2==m)M(x,y) else L(x,y);} fprintf(o,"0 .8 0 RGB .03 W S\n"); //GREEN DASH

DO(m,36){ x=2.+.1*(m+.5); z=x; c=F45E(.5+F45L(z));y=Re(c);if(m==0)M(x,y) else L(x,y);} fprintf(o,"0 .4 0 RGB .01 W S\n"); //GREEN DARK //ok

DO(m,56){ x=4.-.123*(m+.5); z=x; c=F21E(.1+F21L(z));y=Re(c);if(m/2*2==m)M(x,y) else L(x,y); } fprintf(o,".8 0 0 RGB .03 W S\n"); //RED DASH // ok

DO(m,41){ x=2.+.1*(m+.5); z=x; c=F45E(.1+F45L(z));y=Re(c);if(m==0)M(x,y) else L(x,y); } fprintf(o,".4 0 0 RGB .01 W S\n"); //RED DARK //ok

DO(m,46){ x=4.-.12*(m+.5); z=x; c=F21E(-.1+F21L(z));y=Re(c);if(m/2*2==m)M(x,y) else L(x,y); } fprintf(o,".8 0 0 RGB .03 W S\n"); //RED DASH bottom

DO(m,44){ x=2.+.1*(m+.5); z=x; c=F45E(-.1+F45L(z));y=Re(c);if(m==0)M(x,y) else L(x,y); } fprintf(o,".4 0 0 RGB .01 W S\n"); //RED dark

DO(m,36){ x=4.-.12*(m+.5); z=x; c=F21E(-.5+F21L(z));y=Re(c);if(m/2*2==m)M(x,y) else L(x,y); } fprintf(o,"0 .8 0 RGB .03 W S\n"); //GREEN DASH //ok

DO(m,49){ x=2.+.1*(m+.5); z=x; c=F45E(-.5+F45L(z));y=Re(c);if(m==0)M(x,y) else L(x,y); } fprintf(o,"0 .4 0 RGB .01 W S\n"); //GREEN DARK //ok

DO(m,42){ x=4.-.0872*(m+.5); z=x; c=F21E(-.9+F21L(z));y=Re(c);if(m/2*2==m)M(x,y) else L(x,y); } fprintf(o,"0 0 .8 RGB .03 W S\n"); //BLUE DASH

DO(m,60){ x=2.+.1*(m+.5); z=x; c=F45E(-.9+F45L(z));y=Re(c);if(m==0)M(x,y) else L(x,y); } fprintf(o,"0 0 .4 RGB .01 W S\n"); //BLUE DARK

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf e1e14cz.eps"); system( "open e1e14cz.pdf");           //for macintosh getchar; system("killall Preview");  //for macintosh }

Latex generator of labels
\documentclass[12pt]{article} %\paperwidth 472px %\paperheight 800px \paperwidth 428px \paperheight 756px \textwidth 704px \textheight 900px \topmargin -120px \oddsidemargin -72px \usepackage{graphics} \usepackage{rotating} \usepackage[usenames]{color} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \parindent 0pt \pagestyle{empty} \begin{document} \newcommand \scalr { \put(370,766){\sx{6}{$y$}} \put(370,583){\sx{6}{$4$}} \put(370,383){\sx{6}{$2$}} \put(370,183){\sx{6}{$0$}} \put( 600,150){\sx{6}{$2$}} \put( 799,150){\sx{6}{$4$}} \put( 998,150){\sx{6}{$6$}} \put(1198,150){\sx{6}{$8$}} \put(1380,150){\sx{6}{$x$}} } \sx{.3}{\begin{picture}(1400,850) \put(0,-10){\ing{e1e14az}}\scalr \put(0,620){\sx{10}{ \color{white}{\rule{24pt}{12pt}} }} \put(10,640){\sx{10}{ $b\!=\!\mathrm e$}} \put( 810,666){\rot{51}\sx{4.4}{$n\!=\!0.1$}\ero} \put( 925,696){\rot{45}\sx{4.8}{$n\!=\!0$}\ero} \put( 934,600){\rot{37}\sx{4.4}{$n\!=\!-0.1$}\ero} \put(1230,570){\rot{16}\sx{4.4}{$n\!=\!-0.5$}\ero}% \put(1220,436){\rot{8}\sx{4.5}{$n\!=\!-0.9$}\ero}% \put(1222,374){\rot{ 6}\sx{4.6}{$n\!=\!-1$}\ero}% \put(1230,230){\rot{ 4}\sx{4.6}{$n\!=\!-2$}\ero} \put(446,525){\rot{81}\sx{4.3}{$y\!=\!\exp(\mathrm e^x)$}\ero} \put(499,485){\rot{75}\sx{4.4}{$y\!=\!\exp(x)$}\ero} \put(595,485){\rot{67}\sx{4.4}{$y\!=\!\sqrt{\exp}(x)$}\ero} \put(840,426){\rot{19}\sx{4.6}{$y\!=\!\sqrt{\ln}(x)$}\ero} \put(890,330){\rot{ 8}\sx{4.6}{$y\!=\!\ln(x)$}\ero} \put(890,232){\rot{ 6}\sx{4.6}{$y\!=\!\ln(\ln(x))$}\ero} \put(14,315){\rot{1}\sx{4.8}{$n\!=\!2$}\ero} \put(14,210){\rot{4}\sx{4.8}{$n\!=\!1$}\ero} \put(14,155){\rot{2}\sx{4.7}{$n\!=\!0.9$}\ero} \put(20, 95){\rot{ 3}\sx{4.7}{$n\!=\!0.5$}\ero} \put(234,2){\rot{45}\sx{5.5}{$y\!=\!x$}\ero} \put(450, 0){\rot{69}\sx{4.7}{$n\!=\!-1$}\ero} \put(550,-2){\rot{62}\sx{4.5}{$n\!=\!-2$}\ero} \end{picture}}

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\end{document}