File:AcosqqplotT.png

Graphic of function ArcCosqq (or acosqq) defined with
 * $\text{acosqq}(z)=\text{acosq}(z)\, \tan\!\!\big( \text{acosq}(z) \big)$

through functions acosq (or ArcCosq); it is expressed as;
 * $\text{acosq}(z)=\text{acosq}\big( \mathrm e ^{\mathrm i \pi /4}\, z \big)$

which is ArcCosc of argument rotated for the phase quarter of $\pi$. Namely this phase appear in the application for the atom optics, for channelling of particle between absorbing walls.

acosc or ArcCosc is inverse funciotn of cosc,
 * $\displaystyle \text{cosc}(z)=\frac{\cos(z)}{z}$

Use of function acosqq
acoscqq expresses the decay constant of mode guided between absorbing waves, see the special article Absorbing Schroedinger for the details.

C++ generator of curves
Files ado.cin and acosc.cin should be loaded into the working directory in order to compile the C++

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.)


 * 1) include "ado.cin"


 * 1) include "acosc.cin"

z_type acosq(z_type z){ z_type c=z*exp(I*M_PI/4.); c=acosc(c); return c;}

z_type acosqq(z_type z){ z_type c=z*exp(I*M_PI/4.); c=acosc(c); return c*tan(c);}

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("acosqqplot.eps","w");ado(o,420,460); fprintf(o,"10 110 translate\n 100 100 scale\n"); for(m=0;m<5;m++){M(m,-1)L(m,3)} for(n=-1;n<4;n++){M(0,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,381){x=0.198+.01*m; y=Re(acosqq(x)); if(m==0)M(x,y)else L(x,y); printf("%6.3f %6.3f\n",x,y); } fprintf(o,"1 setlinejoin 1 setlinecap .011 W 0 .6 0 RGB S\n"); DO(m,330){x=0.73+.01*m; y=Im(acosqq(x)); if(m==0)M(x,y)else L(x,y) } fprintf(o,"1 setlinejoin 1 setlinecap .01 W .7 0 .7 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 acosqqplot.eps"); system(   "open acosqqplot.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 acosqqplot.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 812pt % \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}(140,444) % %\put(4,6){\ing{sazaecon}} % \put(16,402){\sx{2.5}{$y$}} % \put(16,308){\sx{2.4}{\bf 2}} % %\put(422,269){\sx{2.4}{$\pi/2$}} % \put(16,208){\sx{2.4}{\bf 1}} % %\put(16, 262){\sx{2.2}{Wakame}} % \put(16,108){\sx{2.4}{\bf 0}} % %\put(16, 75){\sx{2.4}{Tarao}} % %\put(200, 118){\sx{2.4}{\bf -2}} % %\put(300, 118){\sx{2.4}{\bf -1}} % \put(108, 118){\sx{2.4}{\bf 1}} % %\put(142,120){\sx{2.4}{\rot{90}Fune\ero}} % \put(208, 118){\sx{2.4}{\bf 2}} % %\put(302,118){\sx{2.5}{\rot{90}Sazae\ero}} % \put(308, 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(400, 119){\sx{2.4}{$x$}} % \put(4,6){\ing{acosqqplot}} % \put(50,270){\sx{2.52}{\rot{0}$y\!=\!\Re(\mathrm{acosqq}(x))$\ero}} % \put(148, 60){\sx{2.52}{\rot{0}$y\!=\!\Im(\mathrm{acosqq}(x))$\ero}} % \end{picture} % } % \end{document} %