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

using namespace std;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)

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"

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");
 * 1) include "ado.cin"
 * 2) define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
 * 3) define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);

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