File:LogQ2mapT2.png

Complex map of logarithm to base $b\!=\!\sqrt{2}$;

$u\!+\!\mathrm v=\log_b(x\!+\!\mathrm i u)$

The cut of the range of holomorphism is marked with dashed line. Lines $u\!=\!1$, $u\!=\!2$, $u\!=\!4$, $u\!=\!6$ pass through the integer values at the real axis.

Line $u\!=\!7$ tries to pass through the points ($x\!=\!\pm 8$, $y\!=\!\pm 8$), but these points are already near the corners of the mesh.

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=201,M1=M+1;
int N=203,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("LogQ2map.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]=2.012* sinh( log(8.)*(-1.+.01*(m-.5)) );
DO(n,101) Y[n]=2.012* sinh( log(8.)*(-1.+.01*(n-.5)) );
          Y[101]=-0.001;
          Y[102]= 0.001;
for(n=103;n<N1;n++) Y[n]=2.01*sinh( log(8.)*(-1.+.01*(n-.5-2)) );
DO(n,N1) printf("%3d %8.5f\n",n,Y[n]);
getchar();
//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=log(z)/log(B);
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=1.;q=.5;
for(m=-9;m<9;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<7;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<10;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<10;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<8;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) {
y=0.;
    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-.12,y) L(x-.24,y)} fprintf(o,".06 W 1 .5 0 RGB S\n");
   for(m=2;m<81;m+=4) {x=-.1*m; M(x-.12,y) L(x-.24,y)} fprintf(o,".06 W 0 .5 1 RGB S\n");
//                           }

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
      system("epstopdf LogQ2map.eps");    
      system(    "open LogQ2map.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 LogQ2map.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 850pt % <br> \paperheight 833pt % <br> \topmargin -104pt % <br> \oddsidemargin -40pt % <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{5}{\begin{picture}(162,162) % <br> \put(1,1){\ing{LogQ2map}} % <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(095,139.2){\sx{.7}{\rot{80}$v\!=\!4$\ero}} % <br> \put(114,132.2){\sx{.7}{\rot{60}$v\!=\!3$\ero}} % <br> \put(129,118.6){\sx{.7}{\rot{40}$v\!=\!2$\ero}} % <br> \put(138,100.2){\sx{.7}{\rot{20}$v\!=\!1$\ero}} % <br> \put(004,084.2){\sx{.66}{$v\!=\!9$}} % <br> \put(141,080.2){\sx{.7}{$v\!=\!0$}} % <br> \put(003,076){\sx{.66}{$v\!=\!-9$}} % <br> %\put(140,059){\sx{.7}{\rot{-20}$v\!=\!-1$\ero}} % <br> \put(136,061){\sx{.66}{\rot{-20}$v\!=\!-1$\ero}} % <br> \put(125,044){\sx{.66}{\rot{-40}$v\!=\!-2$\ero}} % <br> \put(109.2,032){\sx{.66}{\rot{-60}$v\!=\!-3$\ero}} % <br> %\put(091.2,025){\sx{.66}{\rot{-79}$v\!=\!-4$\ero}} % <br> % <br> \put(80,051){\sx{.66}{\rot{11}$u\!=\!3$\ero}} % <br> \put(83,039.6){\sx{.66}{\rot{11}$u\!=\!4$\ero}} % <br> \put(86,023.3){\sx{.66}{\rot{11}$u\!=\!5$\ero}} % <br> \put(90,000.2){\sx{.66}{\rot{11}$u\!=\!6$\ero}} % <br> \end{picture}} % <br> \end{document} % <br> %