File:Acosc1plotT.png

Plot of Acosc1, which is first branch of inverse function of cosc$(x)=\cos(x)/x$.

Description

 * $y=\text{acosc}_1(x)$ is shown with thick red line.

The two asymptotics are shown with thin lines:


 * $\displaystyle y= \mathrm R_3(x) =

\mathrm{Sazae}_1 - \sqrt{ \frac{2}{\mathrm{Tarao}_1 } (\mathrm{Tarao}_1\!-\!x) } + \frac{2\, (\mathrm{Tarao}_1 \!-\! x )}{3~ \mathrm{Sazae}_1~ \mathrm{Tarao}_1}$ which approximates the function in vicinity of $\mathrm{Tarao}_1$, and
 * $\displaystyle y= \mathrm L_3(x) =

\mathrm{Sazae}_0 + \sqrt{ \frac{-2}{\mathrm{Tarao}_0} (x-\mathrm{Tarao}_0) } - \frac {2\, (x\!-\!\mathrm{Tarao}_0)} {3~ \mathrm{Tarao}_0 ~ \mathrm{Sazae}_0} $ which aprocimates the function in vicinity of Tarao$=\mathrm{Tarao}_0$. These asymptotics are used in the numerical implementation acosc1 available at acosc1.cin.

At the bottom part of the figure, function ArcCosc=acosc is also plotted and compared to its approximation


 * $\displaystyle y=\mathrm D_3 (x)=

\mathrm{Sazae}_0 - \sqrt{ \frac{-2}{\mathrm{Tarao}_0} (x-\mathrm{Tarao}_0) } - \frac {2\, (x\!-\!\mathrm{Tarao}_0)} {3~ \mathrm{Tarao}_0 ~ \mathrm{Sazae}_0} $

Constants Sazae are positive solutions $x$ of equation $\text{cosc}'(x)=0$.

Constants Tarao are defined with
 * $\mathrm{Tarao}_n=\mathrm{cosc}(\mathrm{Sazae}_n)$

In particular,
 * $\!\!\! \mathrm{Sazae}_0=~$Sazae$~ \approx 2.798386045783887$, $

\mathrm{Tarao}_0=~$Tarao$~\approx\! −0.33650841691839534 $


 * $\!\!\! \mathrm{Sazae}_1=~$Sazae$_1\!\approx 6.1212504668980685~$,  $

\mathrm{Tarao}_1\!=~$Tarao$_1\!\approx 0.161228034325064 $

C++ generator of curves
// Files ado.cin and acosc1.cin should be loaded to the working directory in order to compile 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"
 * 5) include "acosc1.cin"


 * 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",);

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("acosc1plot.eps","w");ado(o,230,670); fprintf(o,"60 10 translate\n 100 100 scale\n"); for(m=0;m<2;m++){M(m,0)L(m,6.5)} for(n=0;n<7;n++){M(-.5,n)L(1.5,n)} fprintf(o,"2 setlinecap .006 W 0 0 0 RGB S\n"); for(m=-1;m<2;m++){M(.5+m,0)L(.5+m,6.5)} for(n=0;n<7;n++){M(-.5,n+.5)L(1.5,n+.5)} fprintf(o,"2 setlinecap .002 W 0 0 0 RGB S\n"); fprintf(o,"2 setlinecap 1 setlinejoin \n"); // /* M(Tarao,Sazae) DO(m,200){x=-.33+.01*(m+.5); z=x; DB t=x-Tarao0, u=sqrt(t); y=Sazae + u*(2.437906425896532+t*0.50093301330428) + t*(0.707954233164988+t*0.57144599327344); L(x,y);} fprintf(o,".06 W 0 1 0 RGB S\n"); */ // M(Tarao0,Sazae0) DO(m,200){x=Tarao+(Tarao1-Tarao)*.02*(m+.5); y=Re(acosc(x)); L(x,y) } fprintf(o,"1 setlinejoin 1 setlinecap .007 W 0 0 1 RGB S\n"); // M(Tarao0,Sazae0) DO(m,100){x=Tarao+(Tarao1-Tarao)*.01*(m+.5); y=Re(acosc1(x)); L(x,y) } L(Tarao1, Sazae1) fprintf(o,"1 setlinejoin 1 setlinecap .02 W 1 0 0 RGB S\n"); // M(Tarao,Sazae) DO(m,70){x=-.33+.01*(m+.5); z=x; y=Sazae + sqrt((-2./Tarao)*(x-Tarao)) - 2./(3.*Sazae*Tarao)*(x-Tarao) ;   L(x,y);} fprintf(o,".005 W 0 0 0 RGB S\n"); // M(Tarao,Sazae) DO(m,200){x=-.33+.01*(m+.5); z=x; y=Sazae - sqrt((-2./Tarao)*(x-Tarao)) - 2./(3.*Sazae*Tarao)*(x-Tarao) ;   L(x,y);} fprintf(o,".005 W 0 0 0 RGB S\n"); // M(Tarao1,Sazae1) DO(m,70){x=Tarao1-.01*(m+.5); DB t=Tarao1-x; DB u=sqrt((2./Tarao1)*t); y=Sazae1 - u + 1./3./Sazae1 * t; L(x,y)} fprintf(o,".005 W 0 0 0 RGB S\n"); // M(-.1,3.*M_PI/2) L(.1,3.*M_PI/2) M(-.1,M_PI/2)L(.1,M_PI/2) M(Tarao,0)L(Tarao,Sazae)L(0,Sazae) M(Tarao1,0)L(Tarao1,Sazae1)L(0,Sazae1) fprintf(o,"2 setlinecap .003 W 0 0 0 RGB S\n"); // fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf acosc1plot.eps"); system(   "open acosc1plot.pdf"); getchar; system("killall Preview");//for mac }

Latex generator of labels
% File acosc1plot.pdf ahould be generated with the code above on order to compile the Latex document below:

% \documentclass[12pt]{article} % \paperheight 838px % \paperwidth 844px % \textwidth 1294px % \textheight 1200px % \topmargin -80px % \oddsidemargin -80px % \usepackage{graphics} % \usepackage{rotating} % \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \newcommand \rmi {\mathrm{i}} % \begin{document} % \newcommand \zoomax { % \put(16,820){\sx{4.4}{$y$}} % \put(16,630){\sx{4}{$2$}} % \put(16,430){\sx{4}{$0$}} % \put(-4, 230){\sx{4}{$-\!2$}} % \put(220, 5){\sx{4}{$-\!2$}} % \put(443, 5){\sx{4}{$0$}} % \put(643, 5){\sx{4}{$2$}} % \put(831,6){\sx{4}{$x$}} % } % \parindent 0pt % %\sx{8}{\begin{picture}(86,86) \put(0,0){\ing{b271t0}} % \begin{picture}(816,816) % \put(40,30){\sx{10}{\ing{1h}}} % \put(40,30){\sx{10}{\ing{1b}}} % \zoomax % \end{picture} % \end{document} % %