File:AciplotTa.png

Plot of ArcCip of real argment.

$y=\mathrm{ArcCip}(x)$ in the $x$, $y$ plane.

Acip.cin

This is preliminary version of the acip.cin file. It looks ugly, but it works!

z_type Cip(z_type z) {return cos(z)/z;}
z_type Cipp(z_type z) {return (-sin(z) - cos(z)/z)/z ;}
z_type ACip0(z_type z) {z_type t=1./(z*z);
       return (1.
       +t*(-1./2.
       +t*(13./24.
       +t*(-541./720.
       +t*(9509./8064.
       +t*(-7231801./3628800.
       +t*(1695106117./479001600.
       +t*(-567547087381./87178291200.)))))))
       )/z ; }

z_type ACip1(z_type z){int n; z_type c=ACip0(z);
       c+=.1*(z-Cip(c))/Cipp(c) ;
       c+=.2*(z-Cip(c))/Cipp(c) ;
       c+=.3*(z-Cip(c))/Cipp(c) ;
       c+=.4*(z-Cip(c))/Cipp(c) ;
       c+=.5*(z-Cip(c))/Cipp(c) ;
       c+=.7*(z-Cip(c))/Cipp(c) ;
       DO(n,8) c+=(z-Cip(c))/Cipp(c) ;
       return c;
       } // ACip0 does not know about the singulatiry.

z_type acos(z_type z){
if(Im(z)<0){if(Re(z)>=0){return -I*log( z + sqrt(z*z-1.) );} 
                    else{return -I*log( z - sqrt(z*z-1.) );}}
            if(Re(z)>=0){return  I*log( z + sqrt(z*z-1.) );} 
                   else {return  I*log( z - sqrt(z*z-1.) );} }

z_type ACipb(z_type z){ z_type t = 0.33650841691839534 + z ; z_type u=sqrt(t);
return  
  2.798386045783887
+u*(-2.437906425896532
+u*(0.7079542331649882 
+u*(-.5009330133042798
+u*(0.5714459932734446 
)))); }

z_type ACip3(z_type z){int n; z_type c=ACipb(z);
       c+=.2*(z-Cip(c))/Cipp(c) ;
       c+=.5*(z-Cip(c))/Cipp(c) ;
       DO(n,8) c+=(z-Cip(c))/Cipp(c) ;
//        DO(n,8) c=acos(c*z) ;
//        DO(n,8) c=cos(c)/z ;
       return c;
       } 

z_type ACip4(z_type z){int n; z_type z1=z_type(0,1.62134794610324);
       z_type t=z-z1; 
       z_type c;
       c=-z1+2.*sqrt(I*t);
//     c=z1+t*(-2.+t*z_type(0,4.) );
       c+=.2*(z-Cip(c))/Cipp(c) ;
       c+=.5*(z-Cip(c))/Cipp(c) ;
       DO(n,8) c+=.9*(z-Cip(c))/Cipp(c) ;
       return c;
       }

z_type ACip5(z_type z){int n;
       if(Re(z)>0 && abs(z)>1.2) return ACip1(z);
       if(Re(z)>-2 && fabs(Im(z))<1.) return ACip3(z);
       if(Im(z)>0) return ACip4(z);
       return conj(ACip4(conj(z)));
}

C++ generator of curve

Fiels ado.cin and acip.cin should be loaded at the working directory for compilation of 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 "ado.cin"
#include "acip.cin"
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
#define S(x,y) fprintf(o,"S\n",);
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
DB y1= 1.1996786402577337;
DB zL= 2.798386045783887;
DB fL= -0.33650841691839534;
FILE *o;o=fopen("aciplot.eps","w");ado(o,320,330);
fprintf(o,"60 10 translate\n 100 100 scale\n");
for(m=0;m<4;m++){M(m,0)L(m,3)}
for(n=0;n<4;n++){M(-.5,n)L(2.5,n)}
fprintf(o,"2 setlinecap .01 W 0 0 0 RGB S\n");
for(m=-1;m<3;m++){M(.5+m,0)L(.5+m,3)}
for(n=0;n<4;n++){M(-.5,n+.5)L(2.5,n+.5)}
fprintf(o,"2 setlinecap .005 W 0 0 0 RGB S\n");
M(fL,zL)
DO(m,2901){ x=-.336+.001*m; z=x; y=Re(ACip5(z)); L(x,y) }
fprintf(o,"1 setlinejoin 1 setlinecap .02 W .7 0 0 RGB S\n"); p=1.8;q=.7;
M(fL,0)L(fL,zL)L(0,zL)
fprintf(o,"2 setlinecap .003 W 0 0 0 RGB S\n"); 
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
      system("epstopdf aciplot.eps");
      system(    "open aciplot.pdf");
      getchar(); system("killall Preview");//for mac
}

Lated generator of lables

File aciplot.pdf ahould be generated with the code above before to compile the Latex document below.

% Copyleft 2012 by Dmitrii Kouznetsov %<br> \documentclass[12pt]{article} %<br> \usepackage{geometry} %<br> \usepackage{graphicx} %<br> \usepackage{rotating} %<br> \paperwidth 610pt %<br> \paperheight 610pt %<br> \topmargin -92pt %<br> \oddsidemargin -106pt %<br> \textwidth 900pt %<br> \textheight 900pt %<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> \parindent 0pt \sx{2}{ \begin{picture}(340,310) %<br> %\put(4,6){\ing{acosplot}} %<br> \put(4,6){\ing{aciplot}} %<br> \put(66,306){\sx{2}{$y$}} %<br> \put(42,290){\sx{2}{$f_0$}} %v \put(66,210){\sx{2}{\bf 2}} %<br> \put(66,110){\sx{2}{\bf 1}} %<br> \put(21, 21){\sx{2.1}{$z_0$}} %<br> \put(58, 19){\sx{2}{\bf 0}} %<br> \put(158, 19){\sx{2}{\bf 1}} %<br> \put(258, 19){\sx{2}{\bf 2}} %<br> \put(300, 21){\sx{2.1}{$x$}} %<br> \put(168,94){\sx{1.6}{\rot{-16}$y\!=\!\mathrm{ACip}(x)$\ero}} \end{picture} %<br> } %<br> \end{document}