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:

using namespace std; typedef complex z_type; 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; 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 }
 * 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) define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x, 0.+y);
 * 2) define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x, 0.+y);

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