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 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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