File:YulyPlot.png

Explicit plot of Yylya function,

$f=\mathrm{Yulya}_a(x)$ for $a=0 .. 1$ versus $x$.

Generator of curves

The function ado.cin is necessary to C++ the paragraph below

1. 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;

DB Yulya(DB a, DB x){ DB p=a+x; DB m=a-x;
 return p/sqrt(1.-p*p)-m/sqrt(1.-m*m);}

DB Yulyap(DB a, DB x){ DB p,m; p=a+x; m=a-x; p=1.-p*p; m=1.-m*m;
 return 1./(p*sqrt(p))+1./(m*sqrt(m));}

DB ArcYulya0(DB a, DB f){
if(a<0.) return ArcYulya0(-a,f);
if(f<0.) return -ArcYulya0(a,-f);
DB m=1.-a; DB p=1.+a;
 return m*f / sqrt( 4./(m*p*p*p) + f*f );}

DB ArcYulya2(DB a, DB f){ int n; DB y=ArcYulya0(a,f);
 DO(n,7) y+=(f-Yulya(a,y))/Yulyap(a,y);        return y;}
//#include "yulya.cin"

#include "ado.cin"
#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, a;
FILE *o;o=fopen("yuly.eps","w");ado(o,208,508);
fprintf(o,"104 204 translate\n 100 100 scale\n");

for(m=-10;m<22;m+=5){  M(.1*m,-2)L(.1*m,3)}
for(n=-20;n<41;n+=5){  M(-1,.1*n)L(1,.1*n)}
fprintf(o,".002 W 0 0 0 RGB S\n");

for(m=1;m<10;m++){  a=.1*m;
for(n=0;n<51;n++){y=-2.+.1*n; x=ArcYulya2(a,y); if(n==0)M(x,y) else L(x,y)
               }}
fprintf(o,".003 W 0 .5 0 RGB S\n");
//a=.15;  for(n=0;n<51;n++){y=-2.+.1*n; x=ArcYulya2(a,y); if(n==0)M(x,y) else L(x,y) }
//a=.05;  for(n=0;n<51;n++){y=-2.+.1*n; x=ArcYulya2(a,y); if(n==0)M(x,y) else L(x,y) }

a=.0000001; for(n=0;n<51;n++){y=-2.+.1*n; x=ArcYulya2(a,y); if(n==0)M(x,y) else L(x,y) }

fprintf(o,".001 W .5 0 0 RGB S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
       system("epstopdf yuly.eps");
       system(    "open yuly.pdf"); //these 2 commands may be specific for macintosh
getchar(); system("killall Preview");// if run at another operational sysetm, may need to modify
}
//End of generator of yuly.pdf

Generator of labels

% Copyleft 2011 by Dmitrii Kouznetsov

\documentclass[12pt]{article} %

\usepackage{geometry} %

\usepackage{graphicx} %

\usepackage{rotating} %

\paperwidth 1052pt %

\paperheight 2556pt %

\topmargin -110pt %

\oddsidemargin -100pt %

\textwidth 1200pt %

\textheight 5200pt %

\pagestyle {empty} %

\newcommand \sx {\scalebox} %

\newcommand \rot {\begin{rotate}} %<br>
 \newcommand \ero {\end{rotate}} %

\newcommand \ing {\includegraphics} %

\begin{document} %<br>
 \sx{5}{ \begin{picture}(208,509) %<br>
 \put(3,2){\ing{yuly}} %<br>
 \put(1,501){\sx{.8}{$f$}} %<br>
 \put(1,404){\sx{.8}{$2$}} %<br>
 \put(1,304){\sx{.8}{$1$}} %<br>
 \put(1,204){\sx{.8}{$0$}} %<br>
 \put(-3,104){\sx{.8}{$-\!1$}} %<br>
 %\put(-3, 4){\sx{.6}{$-2$}}  %<br>
 \put(49,-1){\sx{.8}{$-0.5$}} %<br>
 \put(106,-1){\sx{.8}{$0$}} %<br>
 \put(153,-1){\sx{.8}{$0.5$}} %<br>
 \put(203,0){\sx{.8}{$x$}} %<br>
 \put(107,480){\rot{90} \sx{.66}{$a\!=\!1$} \ero} %<br>
 \put(114,480){\rot{89} \sx{.66}{$a\!=\!0.9$} \ero} %<br>
 \put(123,480){\rot{89} \sx{.66}{$a\!=\!0.8$} \ero} %<br>
 \put(132,480){\rot{88} \sx{.66}{$a\!=\!0.7$} \ero} %<br>
 \put(141,480){\rot{88} \sx{.66}{$a\!=\!0.6$} \ero} %<br>
 \put(151,480){\rot{87} \sx{.66}{$a\!=\!0.5$} \ero} %<br>
 \put(160,480){\rot{87} \sx{.66}{$a\!=\!0.4$} \ero} %<br>
 \put(169,480){\rot{86} \sx{.66}{$a\!=\!0.3$} \ero} %<br>
 \put(178,480){\rot{86} \sx{.66}{$a\!=\!0.2$} \ero} %<br>
 \put(185,480){\rot{85} \sx{.66}{$a\!=\!0.1$} \ero} %<br>
 \put(194,480){\rot{84} \sx{.66}{$a\!=\!0$} \ero} %<br>
 \end{picture} %

} %

\end{document}  %
 

References