File:AcosqplotT100.png

Graphic of function Acosq, defined through ArcCosc with
 * $ \mathrm{acosq}(z)=\mathrm{acosc}\left( \mathrm e^{\mathrm i \pi/4}\, z\right)$

for real values of $z$.

$y=\Re(\mathrm{acosq}(x))$ is shown with blue line.

$y=\Im(\mathrm{acosq}(x))$ is shown with red line.

C++ implementation of ArcCosc
The code belot should be loaded in the working directory for complation of the C++ generator of curves.

z_type cosc(z_type z) {return cos(z)/z;} z_type cosp(z_type z) {return (-sin(z) - cos(z)/z)/z ;} z_type cohc(z_type z) {return cosh(z)/z ;} z_type cohp(z_type z) {return (sinh(z)-cosh(z)/z)/z ;}

z_type acoscL(z_type z){ int n; z_type s,q; z*=-I; q=I*sqrt(1.50887956153832-z); s=q*1.1512978931181814 + 1.199678640257734; DO(n,6) s+= (z-cohc(s))/cohp(s); return -I*s; }

z_type acoscR(z_type z) {int n; z_type s= (1.-0.5/(z*z))/z; DO(n,5) s+=(z-cosc(s))/cosp(s); return s;}

z_type acoscB(z_type z){ z_type t=0.33650841691839534+z, u=sqrt(t), s; int n; s=   2.798386045783887 +u*(-2.437906425896532 +u*( 0.7079542331649882 +u*(-0.5009330133042798 +u*( 0.5714459932734446   )))); DO(n,6) s+=(z-cosc(s))/cosp(s); return s; }

z_type acosc(z_type z){ DB x1=-0.33650841691839534, x=Re(z), y=Im(z), yy=y*y, r=x-x1;r*=r;r+=yy; if(r < 1.8 )    return acoscB(z); r=x+2.;r*=r;r+=yy; if(r>8. && x>=0) return acoscR(z); if(y >= 0) return acoscL(z); return conj(acoscL(conj(z))); }

C++ generator of curves
Files ado.cin and acosc.cin should be loaded in the working directory in order to complie the C++ code below]]

using namespace std; typedef complex z_type;
 * 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"


 * 1) include "acosc.cin"

main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; DB Sazae= 2.798386045783887; // H DB Tarao= -0.33650841691839534; // J FILE *o;o=fopen("acosqplot.eps","w");ado(o,820,460); fprintf(o,"410 110 translate\n 100 100 scale\n"); for(m=-4;m<5;m++){M(m,-1)L(m,3)} for(n=-1;n<4;n++){M(-4,n)L(4,n)} fprintf(o,"2 setlinecap .005 W 0 0 0 RGB S\n"); /* for(m=-4;m<3;m++){M(.5+m,-1)L(.5+m,3)} for(n=-1;n<3;n++){M(-4,n+.5)L(4,n+.5)} fprintf(o,"2 setlinecap .003 W 0 0 0 RGB S\n"); */ DO(m,802){x=-4.01+.01*m; z=x*exp(.25*M_PI*I); y=Re(acosc(z)); if(m==0)M(x,y)else L(x,y) } fprintf(o,"1 setlinejoin 1 setlinecap .01 W 0 0 .8 RGB S\n"); DO(m,802){x=-4.01+.01*m; z=x*exp(.25*M_PI*I); y=Im(acosc(z)); if(m==0)M(x,y)else L(x,y) } fprintf(o,"1 setlinejoin 1 setlinecap .01 W .8 0 0 RGB S\n"); M(-.1,M_PI/2) L( .1,M_PI/2) fprintf(o,".004 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf acosqplot.eps"); system(   "open acosqplot.pdf"); getchar; system("killall Preview");//for mac }
 * 1) define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
 * 3) define S(x,y) fprintf(o,"S\n",);

Latex generator of labels
%File acosq.pdf should be generated with the code above in order to compile the Latex document below.

% % Copyleft 2012 by Dmitrii Kouznetsov % \documentclass[12pt]{article} % \usepackage{geometry} % \usepackage{graphicx} % \usepackage{rotating} % \paperwidth 1612pt % \paperheight 878pt % \topmargin -90pt % \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}(840,444) % \put(4,6){\ing{acosqplot}} % %\put(4,6){\ing{sazaecon}} % \put(416,402){\sx{2.5}{$y$}} % \put(416,308){\sx{2.4}{\bf 2}} % \put(422,269){\sx{2.4}{$\pi/2$}} % \put(416,208){\sx{2.4}{\bf 1}} % %\put(16,160){\sx{1.8}{\bf 2.5}} % %\put(16, 262){\sx{2.2}{Wakame}} % \put(416,108){\sx{2.4}{\bf 0}} % %\put(16, 75){\sx{2.4}{Tarao}} % \put(100, 118){\sx{2.4}{\bf -3}} % \put(200, 118){\sx{2.4}{\bf -2}} % \put(300, 118){\sx{2.4}{\bf -1}} % \put(508, 118){\sx{2.4}{\bf 1}} % %\put(142,120){\sx{2.4}{\rot{90}Fune\ero}} % \put(608, 118){\sx{2.4}{\bf 2}} % %\put(302,118){\sx{2.5}{\rot{90}Sazae\ero}} % \put(708, 118){\sx{2.4}{\bf 3}} % %\put(807, 120){\sx{2.2}{\bf 4}} % %\put(161, 132){\sx{2.8}{$\frac{\pi}{2}$}} % %\put(471, 130){\sx{2.6}{$\frac{3\pi}{2}$}} % \put(780, 118){\sx{2.4}{$x$}} % \put(618,160){\sx{2.52}{\rot{-4}$y\!=\!\Re(\mathrm{acosq}(x))$\ero}} % \put(618, 65){\sx{2.52}{\rot{4}$y\!=\!\Im(\mathrm{acosq}(x))$\ero}} % \end{picture} % } % \end{document} %

Keywords
ArcCosc, ArcCosq, Sazae-san functions, Explicit plot