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:

using namespace std; typedef complex z_type; 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 }
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "ado.cin"
 * 5) include "acip.cin"
 * 6) define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 * 7) define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
 * 8) define S(x,y) fprintf(o,"S\n",);

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