File:CoshcplotT.png

Explicit plot of functions Coshc and Coshc' for positive values of the argument.

Coshc or coshc is elementary function, defined with

$\displaystyle \mathrm{coshc}(z)=\frac{\cosh(z)}{z}$

where $\cosh$ is hyperbolic cosine, id est,

$\cosh(z)=(\mathrm e^z+\mathrm e^{-z})/2$

The derivative of coshc, id est, cosec', can be expressed with

$ \cosh'(z)=\frac{\sinh(z)}{z}-\frac{\cosh(z)}{z^2} = \mathrm{sinhc}(z) - \mathrm{coshc}(z)/z$

Functions $~y\!=\!\mathrm{coshc}(x)~$ and $~y\!=\!\mathrm{coshc}'(x)~$ are shown in the $x$,$y$ plane.

Coshc is related with the cosc function with the simple relations

$\displaystyle \mathrm{coshc}(z) = \mathrm i ~ \mathrm{cosc} ( \mathrm i z )$

$\displaystyle \mathrm{cosc}(z) = \mathrm i ~ \mathrm{coshc} ( \mathrm i z )$

quite analogous to the relation between cos and cosh.

Minimum of $\mathrm{coshc}(x)$ is realized at

$x=H\approx 1.199678640257734 ~$ ; $~\mathrm{coshc}'(H)=0$.

$J=\mathrm{coshc}(H) \approx 1.50887956153832~$ .

These values are marked in the fugure.

C++ generator of curvec

File ado.cin should be loaded in the working directory for the compilation of 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"
z_type Cih(z_type z) {return cosh(z)/z ;}
z_type Cihp(z_type z) {return (sinh(z)-cosh(z)/z)/z ;}
  DB H=1.199678640257734;
  DB J=1.50887956153832;
#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x, 0.+y);
#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x, 0.+y);
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
FILE *o;o=fopen("coshcplot.eps","w");ado(o,220,420);
fprintf(o,"10 210 translate\n 100 100 scale\n");
M(0,-2)L(0,2) M(0,0)L(2,0)
fprintf(o,".01 W 0 0 0 RGB S\n");
M(1,-2)L(1,2)
M(2,-2)L(2,2)
M(0, 2)L(2, 2)
M(0, 1)L(2, 1)
M(0,-1)L(2,-1)
M(0,-2)L(2,-2)
M(H,0)L(H,J)L(0,J)
fprintf(o,".004 W 0 0 0 RGB S\n");
DO(m,148){x=.57+.01*m;y=Re(Cih(x));if(m==0)M(x,y)else L(x,y);}
fprintf(o,".014 W 0 0 0 RGB S\n");
DO(m,142){x=.62+.01*m;y=Re(Cihp(x));if(m==0)M(x,y)else L(x,y);}
fprintf(o,".02 W 1 0 0 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
      system("epstopdf coshcplot.eps");
      system(    "open coshcplot.pdf");            
      getchar(); system("killall Preview");//for mac
}

Latex generator of labels

File coshcplot.pdf should be generated with the C++ code above in order to compile the Latex document below:

\documentclass[12pt]{article} %<br> \include{geometry} %<br> \paperwidth 224pt %<br> \paperheight 410pt %<br> \textwidth 300pt %<br> \textheight 600pt %<br> \topmargin -114pt %<br> \oddsidemargin -70pt %<br> \pagestyle{empty} %<br> \usepackage{graphics} %<br> \usepackage{rotating} %<br> \newcommand \sx \scalebox %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \begin{document} %<br> \parindent 0pt %<br> \begin{picture}(240,420) %<br> \put(8,0){\includegraphics{coshcplot}} %<br> \put(4,404){\sx{1.8}{$y$}} %<br> \put(4,355){\sx{1.8}{$J$}} %<br> \put(4,304){\sx{1.8}{$1$}} %<br> \put(4,204){\sx{1.8}{$0$}} %<br> \put(-4,104){\sx{1.8}{$-\!1$}} %<br> \put( 13,192){\sx{1.8}{$0$}} %<br> \put(114,192){\sx{1.8}{$1$}} %<br> \put(131,192){\sx{1.8}{$H$}} %<br> \put(203,194){\sx{1.9}{$x$}} %<br> %\put(-4,5){\sx{1.8}{$-2$}} \put(150,341){\sx{1.7}{\rot{27}{$\mathrm{Coshc}(x)$}\ero}} %<br> \put(152,234){\sx{1.7}{\rot{47}{$\mathrm{Coshc}'(x)$}\ero}} %<br> \end{picture} %<br> \end{document} %<br> Copyleft 2012 by Dmitrii Kouznetsov