Difference between revisions of "File:ArcTaniaMap.png"

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{{oq|ArcTaniaMap.png|Original file ‎(851 × 841 pixels, file size: 626 KB, MIME type: image/png)}}
⚫
[[Complex map]] of the Arc[[Tania function]],
  
⚫
:$ \mathrm{ArcTania}(z)= z+\ln(z) -1$
  
   
−
$f=\mathrm{ArcTania}(x\!+\!\mathrm{i} y)$ in the $x,y$ plane;
+
\[u+\mathrm iv=\mathrm{ArcTania}(x\!+\!\mathrm{i} y)\]
−
lines $u=\Re(f)=\mathrm{const}$ and
  
−
lines $v=\Im(f)=\mathrm{const}$ are drawn.
  
−
The integer values correspond to thick lines.
  
   
  ⚫
[[Complex map]] of the Arc[[Tania function]],
−
==Generators==
  
  ⚫
\[ \mathrm{ArcTania}(z)= z+\ln(z) -1\]
   
  +
This [[complex map]] is used as Fig.5.3 at page 48 of book «[[Superfunctions]]»
⚫
===[[C++]] generator of curves===
  
  +
<ref>
⚫
Files [[ado.cin]] and [[conto.cin]] are necessary to compile the code below
  
  +
https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas,algorithms,tables,graphics ペーパーバック – 2020/7/28
  +
</ref><ref>https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas, algorithms, tables, graphics. Publisher: [[Lambert Academic Publishing]].
  +
</ref>
  +
<br>
  +
in order to invite the reader to practice with [[complex map]]s and also
  +
to show formalism of [[superfunctions]] with the simple example, that
  +
actually does not require this formalism to deal with.
   
  ⚫
==[[C++]] generator of curves==
  ⚫
//Files [[ado.cin]] and [[conto.cin]] are necessary to compile the code below
  +
<pre>
 
#include <math.h>
  
#include <math.h>
 
#include <stdio.h>
  
#include <stdio.h>
Line 70: Line 77:
 
getchar(); system("killall Preview");//for mac
  
getchar(); system("killall Preview");//for mac
 
}
  
}
  +
</pre>
−  
 
=== [[Latex]] Generator of lables===
  
=== [[Latex]] Generator of lables===
  +
<pre>
−
<poem><nomathjax><nowiki>
  
 
% Gerenator of ArcTaniaMap.png %<br>
  
% Gerenator of ArcTaniaMap.png %<br>
 
% Copyleft 2011 by Dmitrii Kouznetsov %<br>
  
% Copyleft 2011 by Dmitrii Kouznetsov %<br>
Line 139: Line 146:
 
} %<br>
  
} %<br>
 
\end{document}
  
\end{document}
  +
</pre>
−
</nowiki></nomathjax></poem>
  
  +
  +
==References==
  +
{{ref}}
  +
  +
{{fer}}
  +
  +
==Keywords==
  +
<b>«[[ArcTania function]]»</b>,
  +
«[[Complex map]]»,
  +
<b>«[[]]»</b>,
  +
«[[Superfunctions]]»,
  +
<b>«[[Tania function]]»</b>,
   
 
[[Category:Book]]
  
[[Category:Book]]
 
[[Category:BookMap]]
  
[[Category:BookMap]]
 
[[Category:Tania function]]
  
[[Category:Tania function]]
−
[[Category:Coplex maps]]
 +
[[Category:Complex map]]
 
[[Category:Inverse functions]]
  
[[Category:Inverse functions]]

Revision as of 13:17, 17 August 2025


\[u+\mathrm iv=\mathrm{ArcTania}(x\!+\!\mathrm{i} y)\]

Complex map of the ArcTania function, \[ \mathrm{ArcTania}(z)= z+\ln(z) -1\]

This complex map is used as Fig.5.3 at page 48 of book «Superfunctions» [1][2]
in order to invite the reader to practice with complex maps and also to show formalism of superfunctions with the simple example, that actually does not require this formalism to deal with.

C++ generator of curves

//Files ado.cin and conto.cin are necessary 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++)
 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"

 z_type ArcTania(z_type z) {return z + log(z) - 1. ;}

 main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  int M=160,M1=M+1;
  int N=161,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("arctaniacontour.eps","w");ado(o,162,162);
 fprintf(o,"81 81 translate\n 10 10 scale\n");
 DO(m,M1) X[m]=-8.+.1*(m);
 DO(n,80)Y[n]=-8.+.1*n;
        Y[80]=-.033;
        Y[81]= .033;
 for(n=82;n<N1;n++) Y[n]=-8.+.1*(n-1.);
 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,".0009 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=ArcTania(z);
 //     p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.;
        p=Re(c);q=Im(c);        
        if(p>-99. && p<99. &&
 //       (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&
           q>-99. && q<99 //&& fabs(q)> 1.e-19
        )
        {g[m*N1+n]=p;f[m*N1+n]=q;}
                        }}
 fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1.8;q=.7;
 //p=2;q=1;
 for(m=-11;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,".011 W 0 .6 0 RGB S\n");
 for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".011 W .9 0 0 RGB S\n");
 for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".011 W 0 0 .9 RGB S\n");
 for(m=1;m<11;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<11;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=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 0 RGB S\n");
 y= 0.; for(m=0;m<80;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} fprintf(o,".07 W 1 .5 0 RGB S\n");
 y= 0.; for(m=2;m<80;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} fprintf(o,".07 W 0 .5 1 RGB S\n");
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf arctaniacontour.eps");
        system(    "open arctaniacontour.pdf");            
        getchar(); system("killall Preview");//for mac
 }

Latex Generator of lables

 % Gerenator of ArcTaniaMap.png %<br>
 % Copyleft 2011 by Dmitrii Kouznetsov %<br>
 \documentclass[12pt]{article} %<br>
 \usepackage{geometry} %<br>
 \usepackage{graphicx} %<br>
 \usepackage{rotating} %<br>
 \paperwidth 854pt %<br>
 \paperheight 844pt %<br>
 \topmargin -96pt %<br>
 \oddsidemargin -98pt %<br>
 \textwidth 1100pt %<br>
 \textheight 1100pt %<br>
 \pagestyle {empty} %<br>
 \newcommand \sx {\scalebox} %<br>
 \newcommand \rot {\begin{rotate}} %<br>
 \newcommand \ero {\end{rotate}} %<br>
 \newcommand \ing {\includegraphics} %<br>
 \begin{document} %<br>
 \sx{5}{ \begin{picture}(164,165) %<br>
 \put(6,5){\ing{arctaniacontour}} %<br>
 \put(2,162){\sx{.7}{$y$}} %<br>
 \put(2,144){\sx{.6}{$6$}} %<br>
 \put(2,124){\sx{.6}{$4$}} %<br>
 \put(2,104){\sx{.6}{$2$}} %<br>
 %\put(23,100){\sx{.8}{$u\!=\!0$}} %<br>
 \put(2, 84){\sx{.6}{$0$}} %<br>
 \put(59,85){\sx{.6}{\bf cut}} %<br>
% \put(20, 84){\sx{.8}{$v\!=\!0$}} %<br>
 \put(-3,64){\sx{.6}{$-2$}} %<br>
 \put(-3,44){\sx{.6}{$-4$}} %<br>
 \put(-3,24){\sx{.6}{$-6$}} %<br>
 \put( 22,0){\sx{.6}{$-6$}} %<br>
 \put( 42,0){\sx{.6}{$-4$}} %<br>
 \put( 62,0){\sx{.6}{$-2$}} %<br>
 \put( 86,0){\sx{.6}{$0$}} %<br>
 \put(106,0){\sx{.6}{$2$}} %<br>
 \put(126,0){\sx{.6}{$4$}} %<br>
 \put(146,0){\sx{.6}{$6$}} %<br>
 \put(164,0){\sx{.7}{$x$}} %<br>

 \put( 81,  23){\rot{81}\sx{.8}{$u\!=\!0$}\ero}%<br>
 \put( 92,  23){\rot{82}\sx{.8}{$u\!=\!1$}\ero}%<br>
 \put(101,  22){\rot{82}\sx{.8}{$u\!=\!2$}\ero}%<br>
 \put(111,  21){\rot{83}\sx{.8}{$u\!=\!3$}\ero}%<br>
 \put(120,  21){\rot{84}\sx{.8}{$u\!=\!4$}\ero}%<br>

 \put(139,155){\rot{4}\sx{.8}{$v\!=\!8$}\ero}%<br>
 \put(138,146){\rot{4}\sx{.8}{$v\!=\!7$}\ero}%<br>
 \put(138,136){\rot{4}\sx{.8}{$v\!=\!6$}\ero}%<br>
 \put(138,127){\rot{4}\sx{.8}{$v\!=\!5$}\ero}%<br>
 \put(137,118){\rot{4}\sx{.8}{$v\!=\!4$}\ero}%<br>
 \put(136,109){\rot{4}\sx{.8}{$v\!=\!3$}\ero}%<br>
 \put(135,100){\rot{4}\sx{.8}{$v\!=\!2$}\ero}%<br>
 \put(134,  92){\rot{3}\sx{.8}{$v\!=\!1$}\ero}%<br>
 \put(134,  84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}%<br>
 \put(134,  76){\rot{-3}\sx{.8}{$v\!=\!-\!1$}\ero}%<br>
 \put(133,  68){\rot{-5}\sx{.8}{$v\!=\!-\!2$}\ero}%<br>
 \put(134,  59){\rot{-5}\sx{.8}{$v\!=\!-\!3$}\ero}%<br>
 \put(135,  51){\rot{-5}\sx{.8}{$v\!=\!-\!4$}\ero}%<br>
 \put(135,  41){\rot{-5}\sx{.8}{$v\!=\!-\!5$}\ero}%<br>
 \put(135,  32){\rot{-5}\sx{.8}{$v\!=\!-\!6$}\ero}%<br>
 \put(136,  23){\rot{-5}\sx{.8}{$v\!=\!-\!7$}\ero}%<br>
 \put(137,  14){\rot{-5}\sx{.8}{$v\!=\!-\!8$}\ero}%<br>
 \end{picture} %<br>
 } %<br>
 \end{document}

References

  1. ↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics ペーパーバック – 2020/7/28
  2. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.

Keywords

«ArcTania function», «Complex map», «[[]]», «Superfunctions», «Tania function»,

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