File:Frac1zt.jpg
Iterates of function $T(z)=-1/z$
$y=T^n(x)$ is plotted versus $x$ for various real values of number $n$ of iteration.
The non-integer iterates of function $T$ are evaluated using the superfunction
$\displaystyle F(z)=\tan\left(\frac{2}{\pi} z\right)$
and the Abel function
$\displaystyle G(z)=F^{-1}(z)=\frac{2}{\pi} \arctan\left( z\right)$
C++ generator of curves
File ado.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 DO(x,y) for(x=0;x<y;x++)
#define DB double
#include"ado.cin"
DB F(DB n,DB x){return tan( (M_PI/2)*n+ atan(x));}
main(){ FILE *o; int m,n,k; DB x,y,t;
o=fopen("frac1z.eps","w");
ado(o,1002,1002);
#define M(x,y) fprintf(o,"%7.4f %7.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%7.4f %7.4f L\n",0.+x,0.+y);
fprintf(o,"501 501 translate 100 100 scale 2 setlinecap\n");
for(n=-5;n<6;n++) { M(-5,n)L(5,n)}
for(m=-5;m<6;m++) { M(m,-5)L(m,5)}
fprintf(o,".005 W S\n");
M(-5,0)L(5,0)
M(0,-5)L(0,5)
fprintf(o,".02 W S\n");
n=0;DO(m,1001){x=-5.+.01*(m-.5);y=F(2.,x);if(y>-5&&y<5){ if(n==0){M(x,y) n=1;}else L(x,y)} else n=0;} fprintf(o,".03 W 0 1 0 RGB S\n");
n=0;DO(m,1001){x=-5.+.01*(m-.5);y=F(1.,x);if(y>-5&&y<5){ if(n==0){M(x,y) n=1;}else L(x,y)} else n=0;} fprintf(o,".03 W 0 1 0 RGB S\n");
DO(k,20){ t=-2.+.1*k;
n=0;DO(m,1001){x=-5.+.01*(m-.5);y=F(t,x);if(y>-7.2&&y<7.2){ if(n==0){M(x,y) n=1;}else L(x,y)} else n=0;} fprintf(o,".01 W 0 0 0 RGB S\n");
}
fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o);
system("epstopdf frac1z.eps");
system( "open frac1z.pdf");
}
Latex generator of labels
% File frac1z.pdf should be generated with the code above in order to compile the Latex document below.
\documentclass[12pt]{article}
\paperwidth 1006pt
\paperheight 1006pt
\textwidth 1100pt
\textheight 1100pt
\topmargin -108pt
\oddsidemargin -72pt
\parindent 0pt
\pagestyle{empty}
\usepackage {graphics}
\usepackage{rotating}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\begin{document}
\begin{picture}(1004,1004)
\put(479,984){\sx{3}{$y$}}
\put(479,892){\sx{3}{$4$}}
\put(479,792){\sx{3}{$3$}}
\put(479,692){\sx{3}{$2$}}
\put(479,592){\sx{3}{$1$}}
\put(479,492){\sx{3}{$0$}}
\put(459,392){\sx{3}{$-1$}}
\put(459,292){\sx{3}{$-2$}}
\put(459,192){\sx{3}{$-3$}}
\put(459, 92){\sx{3}{$-4$}}
\put( 78,474){\sx{3}{$-4$}}
\put(178,474){\sx{3}{$-3$}}
\put(278,474){\sx{3}{$-2$}}
\put(378,474){\sx{3}{$-1$}}
\put(494,474){\sx{3}{$0$}}
\put(594,474){\sx{3}{$1$}}
\put(694,474){\sx{3}{$2$}}
\put(794,474){\sx{3}{$3$}}
%\put(894,474){\sx{3}{$4$}}
\put(986,475){\sx{3}{$x$}}
%\put(0,0){\ing{fracit05}}
%\put(0,0){\ing{fracit10}}
\put(0,0){\ing{frac1z}}
\put( 62,886){\rot{44}\sx{3}{$n\!=\!1.7$}\ero}
\put(166,774){\rot{44}\sx{3}{$n\!=\!1.6$}\ero}
\put(176,678){\rot{30}\sx{3}{$n\!=\!1.5$}\ero}
\put(180,626){\rot{20}\sx{3}{$n\!=\!1.4$}\ero}
\put(180,588){\rot{16}\sx{3}{$n\!=\!1.3$}\ero}
\put(180,563){\rot{11}\sx{3}{$n\!=\!1.2$}\ero}
%\put(180,530){\rot{12}\sx{3}{$n\!=\!1.1$}\ero}
\put(180,523){\rot{9}\sx{3}{$n\!=\!1$}\ero}
\put(180,441){\rot{7}\sx{3}{$n\!=\!0.5$}\ero}
\put(568,866){\rot{84}\sx{3}{$n\!=\!0.5$}\ero}
\put(620,866){\rot{80}\sx{3}{$n\!=\!0.3$}\ero}
\put(665,866){\rot{78}\sx{3}{$n\!=\!0.2$}\ero}
\put(734,866){\rot{69}\sx{3}{$n\!=\!0.1$}\ero}
\put(840,827){\rot{45}\sx{3}{$n\!=\!0$}\ero}
\put(866,714){\rot{20}\sx{3}{$n\!=\!-0.1$}\ero}
\put(866,643){\rot{12}\sx{3}{$n\!=\!-0.2$}\ero}
\put(866,602){\rot{8}\sx{3}{$n\!=\!-0.3$}\ero}
\put(866,572){\rot{05}\sx{3}{$n\!=\!-0.4$}\ero}
\put(866,529){\rot{04}\sx{3}{$n\!=\!-0.6$}\ero}
\put(866,466){\rot{02}\sx{3}{$n\!=\!-1$}\ero}
\put(866,426){\rot{04}\sx{3}{$n\!=\!-1.2$}\ero}
\put(866,368){\rot{07}\sx{3}{$n\!=\!-1.4$}\ero}
\put(870,320){\rot{09}\sx{3}{$n\!=\!-1.5$}\ero}
\put(872,230){\rot{20}\sx{3}{$n\!=\!-1.6$}\ero}
\put(882, 30){\rot{44}\sx{3}{$n\!=\!-1.7$}\ero}
\end{picture}
\end{document}
References
File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 18:37, 26 August 2013 | | 2,089 × 2,089 (734 KB) | T (talk | contribs) | Iterates of function $T(z)=-1/z$ $y=T^n(x)$ is plotted versus $x$ for various real values of number $n$ of iteration. The non-integer iterates of function $T$ are evaluated using the superfunction $\displaystyle F(z)=\tan\left(\frac{2}{\pi} z\r... |
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