File:Loge1emapT1000.jpg

Complex map of logarithm to base sqrt2, $b=\sqrt{2}\approx 1.41421356237$;

$\ln(b) \approx 0.34657359028$

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

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. using namespace std; typedef complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "conto.cin"

// DB B=sqrt(2.); DB B=exp(1./M_E); DB LB=1./M_E;

int 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("loge1emap.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=log(z)/LB; // c=exp(LB*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=4;q=1.; for(m=-10;m<11;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<11;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<11;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) 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,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 loge1emap.eps"); system(   "open loge1emap.pdf"); printf("r=%16.14f %16.14f\n",r,sqrt(M_PI*M_PI+r*r)); getchar; system("killall Preview"); }

Latex generator of labels
\documentclass[12pt]{article} \paperwidth 170px \paperheight 168px \textwidth 304px \textheight 300px \topmargin -106px \oddsidemargin -72px \usepackage{graphics} \usepackage{rotating} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \parindent 0pt \pagestyle{empty} \begin{document} \parindent 0pt %\put(40,20){\ing{z2itmap}} %\put(1,1){\ing{expe1emap}} \hskip 6pt \begin{picture}(162,161) % \put(1,1){\ing{loge1emap}} % \put(-3,159.9){\sx{.7}{$y$}} \put(-3,140){\sx{.6}{$6$}} \put(-3,120){\sx{.6}{$4$}} \put(-3,100){\sx{.6}{$2$}} \put(-3,80){\sx{.6}{$0$}} \put(-8,60){\sx{.6}{$-2$}} \put(-8,40){\sx{.6}{$-4$}} \put(-8,20){\sx{.6}{$-6$}} \put(16,-4){\sx{.6}{$-6$}} \put(36,-4){\sx{.6}{$-4$}} \put(56,-4){\sx{.6}{$-2$}} \put(81,-4){\sx{.6}{$0$}} \put(101,-4){\sx{.6}{$2$}} \put(121,-4){\sx{.6}{$4$}} \put(141,-4){\sx{.6}{$6$}} \put(159.6,-4){\sx{.7}{$x$}} \put(090,139.2){\sx{.7}{\rot{82}$v\!=\!4$\ero}} \put(109.6,132){\sx{.7}{\rot{62}$v\!=\!3$\ero}} \put(126,118.6){\sx{.7}{\rot{43}$v\!=\!2$\ero}} \put(136,100.6){\sx{.7}{\rot{21}$v\!=\!1$\ero}} \put(039,138){\sx{.66}{\rot{-54}$v\!=\!6$\ero}} \put(022,118.2){\sx{.66}{\rot{-32}$v\!=\!7$\ero}} \put(013,094){\sx{.66}{\rot{-11}$v\!=\!8$\ero}} \put(140,080.2){\sx{.7}{$v\!=\!0$}} \put(010,066){\sx{.66}{\rot{9}$v\!=\!-8$\ero}} \put(020,039.8){\sx{.66}{\rot{33}$v\!=\!-7$\ero}} \put(039.6,021){\sx{.66}{\rot{53}$v\!=\!-6$\ero}} %\put(140,059){\sx{.7}{\rot{-20}$v\!=\!-1$\ero}} \put(133,060.5){\sx{.66}{\rot{-21}$v\!=\!-1$\ero}} \put(123,042.6){\sx{.66}{\rot{-42}$v\!=\!-2$\ero}} \put(105.4,032){\sx{.66}{\rot{-63}$v\!=\!-3$\ero}} %\put(091.2,025){\sx{.66}{\rot{-79}$v\!=\!-4$\ero}} \put(72,066){\sx{.66}{\rot{1}$u\!=\!1$\ero}} \put(72,059){\sx{.66}{\rot{1}$u\!=\!2$\ero}} \put(72,050){\sx{.66}{\rot{1}$u\!=\!3$\ero}} \put(72,036.6){\sx{.66}{\rot{1}$u\!=\!4$\ero}} \put(72,017){\sx{.66}{\rot{1}$u\!=\!5$\ero}} %\put(72,000.2){\sx{.66}{\rot{1}$u\!=\!6$\ero}} \end{picture} \end{document}