Difference between revisions of "File:Sqrt2atemap.jpg"

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[[Complex map]] of [[arctetration]] to base $\sqrt{2}$:
−
Importing image file
  
  +
  +
$u\!+\!\mathrm i v = \mathrm{ate}_{\sqrt{2}}(x\!+\!\mathrm i y)$
  +
  +
  +
Usage: this is figure 16.3 of the book [[Суперфункции]] (2014, In Russian) <ref>
  +
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
  +
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf <br>
  +
http://mizugadro.mydns.jp/BOOK/202.pdf
  +
Д.Кузнецов. Суперфункции. [[Lambert Academic Publishing]], 2014.
  +
</ref>; the English version is in preparation in 2015.
  +
  +
This imahe is used also in figure 2 in the article
  +
<ref>
  +
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br>
  +
http://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf offprint
  +
D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
  +
</ref>.
  +
(top right map)
  +
  +
==Refereces==
  +
<references/>
  +
  +
==[[C++]] generator of the First map==
  +
Files [[ado.cin]],
  +
[[conto.cin]],
  +
[[sqrt2f21e.cin]]
  +
should be loaded in order to compile the 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++)
  +
#include <complex>
  +
typedef std::complex<double> z_type;
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
#include "conto.cin"
  +
// #include "tq2e.cin"
  +
// #include "tq2L.cin"
  +
#include "sqrt2f21L.cin"
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
int M=211,M1=M+1;
  +
int N=201,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("sqrt2f21lma.eps","w"); ado(o,0,0,214,212);
  +
fprintf(o,"112 110 translate\n 10 10 scale\n");
  +
// DB sy=10.1/sinh(N/2./100.);
  +
DO(m,M1) X[m]=-11+.1*(m-.5);
  +
DO(n,N1) Y[n]=-10+.1*(n-.5);
  +
// DO(n,N1) Y[n]=sy*sinh((n-N/2.+.5)/100.);
  +
for(m=-10;m<11;m++) {M(m,-10)L(m,10)}
  +
for(n=-10;n<11;n++) {M( -10,n)L(10,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
  +
/*
  +
fprintf(o,"/adobe-Roman findfont 1 scalefont setfont\n");
  +
for(m=-8;m<0;m+=2) {M(-11.2,m-.3) fprintf(o,"(%1d)s\n",m);}
  +
for(m= 0;m<9;m+=2) {M(-10.7,m-.3) fprintf(o,"(%1d)s\n",m);}
  +
for(m=-8;m<0;m+=4) {M(m-.6,-10.8) fprintf(o,"(%1d)s\n",m);}
  +
for(m= 0;m<9;m+=4) {M(m-.3,-10.8) fprintf(o,"(%1d)s\n",m);}
  +
fprintf(o,"/Times-Italic findfont 1 scalefont setfont\n");
  +
//fprintf(o,"/adobe-italic findfont 1 scalefont setfont\n");
  +
M( 9.6,-10.8) fprintf(o,"(y)s\n");
  +
M(-10.7, 9.5) fprintf(o,"(x)s\n");
  +
// M(-11,0)L(10.1,0) M(0,-11)L(0,10.1) fprintf(o,".01 W 1 0 1 RGB S\n");
  +
// z_type tm,tp,F[M1*N1];;
  +
*/
  +
DO(m,M1)DO(n,N1){ g[m*N1+n]=9999;
  +
f[m*N1+n]=9999;}
  +
DO(m,M1){x=X[m];
  +
DO(n,N1){y=Y[n]; z=z_type(x,y);
  +
if( abs(z-2.)>.2 || abs(z-4.)>.2)
  +
{ c=F21L(z);
  +
p=Re(c); q=Im(c);
  +
if(p>-99 && p<99) g[m*N1+n]=p;
  +
if(q>-99 && q<99 && fabs(q)>1.e-18) f[m*N1+n]=q;
  +
}}}
  +
//p=2; q=1.1;
  +
#include "plofu.cin"
  +
  +
// M(-13.2,0)L(2,0) fprintf(o,".05 W 0 .8 0 RGB S\n");
  +
  +
M(2,0)L(10.1,0)fprintf(o,".05 W 1 1 1 RGB S\n");
  +
DO(n,27){M(2+.3*n,0)L(2+.3*(n+.5) ,0)} fprintf(o,".1 W 0 0 0 RGB S\n");
  +
  +
//M(2,0)L(10.1,0)fprintf(o,".1 W 0 0 0 RGB [.19 .19] 0 setdash S\n");
  +
//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n"); // fails at some printers
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf sqrt2f21lma.eps");
  +
system( "open sqrt2f21lma.pdf"); //for linux
  +
// getchar(); system("killall Preview"); // For macintosh
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of the labels==
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\paperwidth 422px
  +
\paperheight 418px
  +
\textwidth 1394px
  +
\textheight 1300px
  +
\topmargin -94px
  +
\oddsidemargin -76px
  +
\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
  +
  +
\sx{2}{\begin{picture}(204,204) \put(0,0){\ing{sqrt2f21lma}}
  +
\put(6,206){\sx{.8}{$y$}}
  +
\put(6,188){\sx{.8}{$8$}}
  +
\put(6,168){\sx{.8}{$6$}}
  +
\put(6,148){\sx{.8}{$4$}}
  +
\put(6,128){\sx{.8}{$2$}}
  +
\put(6,108){\sx{.8}{$0$}}
  +
\put(-1, 88){\sx{.8}{$-2$}}
  +
\put(-1, 68){\sx{.8}{$-4$}}
  +
\put(-1, 48){\sx{.8}{$-6$}}
  +
\put(-1, 28){\sx{.8}{$-8$}}
  +
\put(24,2){\sx{.8}{$-8$}}
  +
\put(44,2){\sx{.8}{$-6$}}
  +
\put(64,2){\sx{.8}{$-4$}}
  +
\put(84,2){\sx{.8}{$-2$}}
  +
\put(110.5,2){\sx{.8}{$0$}}
  +
\put(130.5,2){\sx{.8}{$2$}}
  +
\put(150.5,2){\sx{.8}{$4$}}
  +
\put(170.5,2){\sx{.8}{$6$}}
  +
\put(190.5,2){\sx{.8}{$8$}}
  +
\put(208.6,2){\sx{.8}{$x$}}
  +
  +
\put(26,198){$v\!=\!0$}
  +
\put(18,152){\rot{-2}$u\!=\!-2$\ero}
  +
\put(26,107.4){$v\!=\!0$} \put(176,108){\bf cut}
  +
\put(18,62){\rot{1}$u\!=\!-2$\ero}
  +
\put(26,17){$v\!=\!0$}
  +
  +
\put(96,172){\rot{20}$v\!=\!0.2$\ero}
  +
\put(106,158){\rot{20}$v\!=\!0.4$\ero}
  +
\put(112,148){\rot{20}$v\!=\!0.6$\ero}
  +
\put(121,139){\rot{20}$v\!=\!1$\ero}
  +
  +
\put(156,167){\rot{37}$u\!=\!-3$\ero}
  +
\put(178,164){\rot{50}$v\!=\!0$\ero}
  +
  +
\put(74,95){\rot{-23}$u\!=\!-1.8$\ero}
  +
\put(87,28){\rot{54}$u\!=\!-2.2$\ero}
  +
\put(113,21){\rot{76}$u\!=\!-2.4$\ero}
  +
\put(132,21){\rot{87}$u\!=\!-2.6$\ero}
  +
\put(137,64){\rot{-79}$u\!=\!-2.8$\ero}
  +
\put(151,51){\rot{-43}$u\!=\!-3$\ero}
  +
  +
\put(175,51){\rot{-48}$v\!=\!0$\ero}
  +
\end{picture}}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
[[Category:Base sqrt2]]
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:C++]]
  +
[[Category:Complex map]]
  +
[[Category:Exp]]
  +
[[Category:Generator]]
  +
[[Category:Latex]]
  +
  +
[[Category:Abel function]]
  +
[[Category:Arctetration]]
  +
[[Category:Superfunction]]
  +
[[Category:Tetration]]

Latest revision as of 08:52, 1 December 2018

Complex map of arctetration to base $\sqrt{2}$:

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


Usage: this is figure 16.3 of the book Суперфункции (2014, In Russian) [1]; the English version is in preparation in 2015.

This imahe is used also in figure 2 in the article [2]. (top right map)

Refereces

  1. ↑ https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
    http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
    http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.
  2. ↑ http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
    http://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf offprint D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.

C++ generator of the First map

Files ado.cin, conto.cin, sqrt2f21e.cin should be loaded in order to compile the code below.


 #include <math.h>
 #include <stdio.h>
 #include <stdlib.h>
 #define DB double
 #define DO(x,y) for(x=0;x<y;x++)
 #include <complex>
 typedef std::complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "conto.cin"
// #include "tq2e.cin"
// #include "tq2L.cin"
#include "sqrt2f21L.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
 int M=211,M1=M+1;
 int N=201,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("sqrt2f21lma.eps","w"); ado(o,0,0,214,212);
 fprintf(o,"112 110 translate\n 10 10 scale\n");
// DB sy=10.1/sinh(N/2./100.);
 DO(m,M1) X[m]=-11+.1*(m-.5);
 DO(n,N1) Y[n]=-10+.1*(n-.5);
// DO(n,N1) Y[n]=sy*sinh((n-N/2.+.5)/100.);
 for(m=-10;m<11;m++) {M(m,-10)L(m,10)}
 for(n=-10;n<11;n++) {M( -10,n)L(10,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
/*
 fprintf(o,"/adobe-Roman findfont 1 scalefont setfont\n");
 for(m=-8;m<0;m+=2) {M(-11.2,m-.3) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<9;m+=2) {M(-10.7,m-.3) fprintf(o,"(%1d)s\n",m);}
 for(m=-8;m<0;m+=4) {M(m-.6,-10.8) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<9;m+=4) {M(m-.3,-10.8) fprintf(o,"(%1d)s\n",m);}
 fprintf(o,"/Times-Italic findfont 1 scalefont setfont\n");
 //fprintf(o,"/adobe-italic findfont 1 scalefont setfont\n");
 M( 9.6,-10.8) fprintf(o,"(y)s\n");
 M(-10.7, 9.5) fprintf(o,"(x)s\n");
// M(-11,0)L(10.1,0) M(0,-11)L(0,10.1) fprintf(o,".01 W 1 0 1 RGB S\n");
// z_type tm,tp,F[M1*N1];;
*/
 DO(m,M1)DO(n,N1){ g[m*N1+n]=9999;
                        f[m*N1+n]=9999;}
 DO(m,M1){x=X[m];
 DO(n,N1){y=Y[n]; z=z_type(x,y);
        if( abs(z-2.)>.2 || abs(z-4.)>.2)
        { c=F21L(z);
                p=Re(c); q=Im(c);
                if(p>-99 && p<99) g[m*N1+n]=p;
                if(q>-99 && q<99 && fabs(q)>1.e-18) f[m*N1+n]=q;
        }}}
//p=2; q=1.1;
 #include "plofu.cin"

// M(-13.2,0)L(2,0) fprintf(o,".05 W 0 .8 0 RGB S\n");

M(2,0)L(10.1,0)fprintf(o,".05 W 1 1 1 RGB S\n");
DO(n,27){M(2+.3*n,0)L(2+.3*(n+.5) ,0)} fprintf(o,".1 W 0 0 0 RGB S\n");

//M(2,0)L(10.1,0)fprintf(o,".1 W 0 0 0 RGB [.19 .19] 0 setdash S\n");
//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n"); // fails at some printers
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf sqrt2f21lma.eps");
        system( "open sqrt2f21lma.pdf"); //for linux
// getchar(); system("killall Preview"); // For macintosh
}

Latex generator of the labels


\documentclass[12pt]{article}
\paperwidth 422px
\paperheight 418px
\textwidth 1394px
\textheight 1300px
\topmargin -94px
\oddsidemargin -76px
\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

\sx{2}{\begin{picture}(204,204) \put(0,0){\ing{sqrt2f21lma}}
\put(6,206){\sx{.8}{$y$}}
\put(6,188){\sx{.8}{$8$}}
\put(6,168){\sx{.8}{$6$}}
\put(6,148){\sx{.8}{$4$}}
\put(6,128){\sx{.8}{$2$}}
\put(6,108){\sx{.8}{$0$}}
\put(-1, 88){\sx{.8}{$-2$}}
\put(-1, 68){\sx{.8}{$-4$}}
\put(-1, 48){\sx{.8}{$-6$}}
\put(-1, 28){\sx{.8}{$-8$}}
\put(24,2){\sx{.8}{$-8$}}
\put(44,2){\sx{.8}{$-6$}}
\put(64,2){\sx{.8}{$-4$}}
\put(84,2){\sx{.8}{$-2$}}
\put(110.5,2){\sx{.8}{$0$}}
\put(130.5,2){\sx{.8}{$2$}}
\put(150.5,2){\sx{.8}{$4$}}
\put(170.5,2){\sx{.8}{$6$}}
\put(190.5,2){\sx{.8}{$8$}}
\put(208.6,2){\sx{.8}{$x$}}

\put(26,198){$v\!=\!0$}
\put(18,152){\rot{-2}$u\!=\!-2$\ero}
\put(26,107.4){$v\!=\!0$} \put(176,108){\bf cut}
\put(18,62){\rot{1}$u\!=\!-2$\ero}
\put(26,17){$v\!=\!0$}

\put(96,172){\rot{20}$v\!=\!0.2$\ero}
\put(106,158){\rot{20}$v\!=\!0.4$\ero}
\put(112,148){\rot{20}$v\!=\!0.6$\ero}
\put(121,139){\rot{20}$v\!=\!1$\ero}

\put(156,167){\rot{37}$u\!=\!-3$\ero}
\put(178,164){\rot{50}$v\!=\!0$\ero}

\put(74,95){\rot{-23}$u\!=\!-1.8$\ero}
\put(87,28){\rot{54}$u\!=\!-2.2$\ero}
\put(113,21){\rot{76}$u\!=\!-2.4$\ero}
\put(132,21){\rot{87}$u\!=\!-2.6$\ero}
\put(137,64){\rot{-79}$u\!=\!-2.8$\ero}
\put(151,51){\rot{-43}$u\!=\!-3$\ero}

\put(175,51){\rot{-48}$v\!=\!0$\ero}
\end{picture}}
\end{document}

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