Difference between revisions of "File:Loge1emapT1000.jpg"

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(Importing image file)
 
 
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[[Complex map]] of [[logarithm]] to [[base sqrt2]], $b=\sqrt{2}\approx 1.41421356237$;
−
Importing image file
  
  +
  +
$\ln(b) \approx 0.34657359028$
  +
  +
$u+\mathrm i v=\log_{\sqrt{2}}(x+\mathrm i y)$
  +
  +
==[[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.
  +
<poem><nomathjax><nowiki>
  +
#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.);
  +
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");
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
<poem><nomathjax><nowiki>
  +
\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) % <br>
  +
\put(1,1){\ing{loge1emap}} % <br>
  +
\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}
  +
</nowiki></nomathjax></poem>
  +
  +
==References==
  +
<references/>
  +
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:Logarithm]]
  +
[[Category:Complex map]]
  +
[[Category:Base e1e]]

Latest revision as of 08:41, 1 December 2018

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.


 #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.);
 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) % <br>
\put(1,1){\ing{loge1emap}} % <br>
\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}

References

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