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]]

#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 "acosc.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 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
}

Latex generator of labels

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

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

Keywords

ArcCosc, ArcCosq, Sazae-san functions, Explicit plot