File:ExpQ2mapT.png

Complex map of exponential to base sqrt2, id est, $b=\sqrt{2}$;

$u\!+\!\mathrm i v=\exp_{\sqrt{2}}(x\!+\!\mathrm i y)$

Note that lines $u\!=\!1$, $u\!=\!2$, $u\!=\!4$, $u\!=\!8$ passthrough the integer values at the real axis.

This function is used as transfer function for the tetration to base sqrt(2) in the illustration of the application of the method of regular iteration to construct the superfunction ^[1].

References

C++ generator of curves

//Files ado.cin and conto.cin should be loaded to the working directory in order to compile the C++ code 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 "conto.cin"

DB B=sqrt(2.);

main(){ int j,k,m,n; DB x,y, p,q, t,r; z_type z,c,d;
r=log(1./(M_E-1.)); printf("r=%16.14f\n",r); 
int M=100,M1=M+1;
int N=400,N1=N+1;
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
char v[M1*N1]; // v is working array
FILE *o;o=fopen("ExpQ2map.eps","w");ado(o,162,162);
fprintf(o,"81 81 translate\n 10 10 scale\n");
// DO(m,M1) {X[m]=-8.+.04*(m);
// DO(m,M1) X[m]=log(exp(-8.)+.02*m*(1.+.3*m));
DO(m,M1) X[m]=4.3* sinh( log(4.)*(-1.+.02*m) );
DO(n,N1) Y[n]=-8.+.04*n;
for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}}
for(n=-8;n<9;n++){     M(  -8,n)L(8,n)}
fprintf(o,".008 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);        
// c=Tania(z); p=Re(c);q=Im(c);  
// c=Shoko(z); p=Re(c);q=Im(c);  
// c=ArcShoka(z); 
// c=Shoka(c); 
c=exp(log(B)*z);
p=Re(c);q=Im(c);  
if(p>-99. && p<99. &&  q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 2 setlinecap\n");  p=10;q=1.;
for(m=-4;m<4;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".01 W 0 .6 0 RGB S\n");
for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".01 W .9 0 0 RGB S\n");
for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".01 W 0 0 .9 RGB S\n");
for(m=1;m<9;m++)  conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".05 W .9 0 0 RGB S\n");
for(m=1;m<9;m++)  conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 .9 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".05 W .6 0 .6 RGB S\n");
for(m=-8;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 0 RGB S\n");
/*
for(y=-2*M_PI;y<7.;y+=2*M_PI) {
    M(0,y)L(-8.1,y) fprintf(o,"0 setlinecap .04 W 1 1 1 RGB S\n");
   for(m=0;m<81;m+=4) {x=-.1*m; M(x,y) L(x-.12,y)} fprintf(o,".06 W 1 .5 0 RGB S\n");
   for(m=2;m<81;m+=4) {x=-.1*m; M(x,y) L(x-.12,y)} fprintf(o,".06 W 0 .5 1 RGB S\n");
                          }
*/
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
      system("epstopdf ExpQ2map.eps");    
      system(    "open ExpQ2map.pdf");
printf("r=%16.14f  %16.14f\n",r,sqrt(M_PI*M_PI+r*r)); 
      getchar(); system("killall Preview");
}

Latex generator of labels

% %<br> % file ExpQ2map.pdf should be generated with the code above in order to compile the Latex document below. % % Copyleft 2012 by Dmitrii Kouznetsov <br> % \documentclass[12pt]{article} % <br> \usepackage{geometry} % <br> \usepackage{graphicx} % <br> \usepackage{rotating} % <br> \paperwidth 1700pt % <br> \paperheight 1666pt % <br> \topmargin -96pt % <br> \oddsidemargin -8pt % <br> \textwidth 1700pt % <br> \textheight 1700pt % <br> \pagestyle {empty} % <br> \newcommand \sx {\scalebox} % <br> \newcommand \rot {\begin{rotate}} % <br> \newcommand \ero {\end{rotate}} % <br> \newcommand \ing {\includegraphics} % <br> \parindent 0pt% <br> \pagestyle{empty} % <br> \begin{document} % <br> \sx{10}{\begin{picture}(162,162) % <br> \put(1,1){\ing{ExpQ2map}} % <br> \put(-2,159.9){\sx{.6}{$y$}} <br> % \put(-2,140){\sx{.6}{$6$}} <br> % \put(-2,120){\sx{.6}{$4$}} <br> % \put(-2,100){\sx{.6}{$2$}} <br> % \put(-2,80){\sx{.6}{$0$}} <br> % \put(-7,60){\sx{.6}{$-2$}} <br> % \put(-7,40){\sx{.6}{$-4$}} <br> % \put(-7,20){\sx{.6}{$-6$}} <br> % \put(-7, 0){\sx{.6}{$-8$}} <br> % \put(-4,-3){\sx{.6}{$-8$}} <br> % \put(16,-3){\sx{.6}{$-6$}} <br> % \put(36,-3){\sx{.6}{$-4$}} <br> % \put(56,-3){\sx{.6}{$-2$}} <br> % \put(81,-3){\sx{.6}{$0$}} <br> % \put(101,-3){\sx{.6}{$2$}} <br> % \put(121,-3){\sx{.6}{$4$}} <br> % \put(141,-3){\sx{.6}{$6$}} <br> % \put(159.6,-3){\sx{.6}{$x$}} <br> % \put(13,125.6){\sx{.7}{$u\!=\!0$}} <br> % \put(13,080.2){\sx{.7}{$v\!=\!0$}} <br> % \put(13,034.8){\sx{.7}{$u\!=\!0$}} <br> % \put(38,119){\sx{.6}{\rot{90}$v\!=\!0.2$\ero}} <br> % \put(58,119){\sx{.6}{\rot{90}$v\!=\!0.4$\ero}} <br> % \put(84.7,119){\sx{.7}{\rot{90}$v\!=\!1$\ero}} <br> % \put(104.7,119){\sx{.7}{\rot{90}$v\!=\!2$\ero}} <br> % \put(116.4,119){\sx{.7}{\rot{90}$v\!=\!3$\ero}} <br> % % <br> % \put(38,73){\sx{.6}{\rot{90}$u\!=\!0.2$\ero}} <br> % \put(58,73){\sx{.6}{\rot{90}$u\!=\!0.4$\ero}} <br> % \put(84.6,74){\sx{.7}{\rot{90}$u\!=\!1$\ero}} <br> % \put(104.6,74){\sx{.7}{\rot{90}$u\!=\!2$\ero}} <br> % \put(116.4,74){\sx{.7}{\rot{90}$u\!=\!3$\ero}} <br> % % <br> % \put(39,24){\sx{.6}{\rot{90}$v\!=\!-0.2$\ero}} <br> % \put(59,24){\sx{.6}{\rot{90}$v\!=\!-0.4$\ero}} <br> % \put(84.6,26){\sx{.7}{\rot{90}$v\!=\!-1$\ero}} <br> % \put(104.6,26){\sx{.7}{\rot{90}$v\!=\!-2$\ero}} <br> % \end{picture}} % <br> \end{document} % <br> %

References