File:ArcYulyaPlot100.png

Contour plot of function inverse of the Yulya function, $\mathrm{ArcYulya}\!=\!\mathrm{ArcYulya}^{-1}$.

Levels of constant $f\!=\!\mathrm{ArcYulya}_a(x)$ are shown in the $x,a$ plane.

Generator of curves
Files conto.cin and ado.cin should be stored in the working directory in order to compile the C++ code 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){ 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){ DB y=ArcYulya0(a,f); y+=(f-Yulya(a,y))/Yulyap(a,y); y+=(f-Yulya(a,y))/Yulyap(a,y); y+=(f-Yulya(a,y))/Yulyap(a,y); y+=(f-Yulya(a,y))/Yulyap(a,y); return y;}

main{ int j,k,m,n; DB x,y, p,q, t; int M=501,M1=M+1; int N=201,N1=N+1; DB X[M1],Y[N1], g[M1*N1], w[M1*N1]; // w is working array. char v[M1*N1]; // v is working array FILE *o;o=fopen("arcyulya.eps","w");ado(o,508,108); fprintf(o,"4 4 translate\n 100 100 scale\n"); DO(m,M1) X[m]=0. +.01*(m-.5); DO(n,N1) Y[n]=0. +.005*(n-.1); for(m=0;m<51;m+=5){    M(.1*m,0) L(.1*m,1)} for(n=0;n<11;n+=5){    M(0,.1*n)L(5,.1*n)} fprintf(o,".006 W 0 0 0 RGB S\n"); DO(m,M1)DO(n,N1){g[m*N1+n]=9999;} DO(m,M1){x=X[m]; //printf("run at x=%6.3f\n",x); DO(n,N1){y=Y[n]; p=x*x+y*y; p=ArcYulya2(y,x); if(p>-85 && p<85) g[m*N1+n]=p; }} fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=2.; conto(o,g,w,v,X,Y,M,N,.01,-p,p);fprintf(o,".005 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,.1,-p,p);fprintf(o,".01 W 0 .5 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,.2,-p,p);fprintf(o,".01 W 0 .5 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,.3,-p,p);fprintf(o,".01 W 0 .5 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,.4,-p,p);fprintf(o,".01 W 0 .5 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,.5,-p,p);fprintf(o,".02 W 0 .4 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,.6,-p,p);fprintf(o,".01 W 0 .5 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,.7,-p,p);fprintf(o,".01 W 0 .5 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,.8,-p,p);fprintf(o,".01 W 0 .5 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,.9,-p,p);fprintf(o,".01 W 0 .5 0 RGB S\n"); fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf arcyulya.eps"); system(   "open arcyulya.pdf"); //these 2 commands may be specific for macintosh getchar; system("killall Preview");// if run at another operational sysetm, may need to modify } //Copyleft 2011 by Dmitrii Kouznetsov
 * 1) include "conto.cin"

Generator of lables
The latex document below includes the arcyulya.pdf generated in the previous section and add the lables. The result is converted to the png format.

\documentclass[12pt]{article} % \usepackage{geometry} % \usepackage{graphicx} % \usepackage{rotating} % \paperwidth 2558pt % \paperheight 540pt % \topmargin -114pt % \oddsidemargin -100pt % \textwidth 2600pt % \textheight 600pt % \pagestyle {empty} % \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \begin{document} % \sx{5}{ \begin{picture}(508,108) % \put(3,2){\ing{arcyulya}} % \put(1,103){\sx{.7}{$a$}} % \put(-2,54){\sx{.6}{$0.5$}} % \put(0,4){\sx{.6}{$0$}} % \put(5,0){\sx{.6}{$0$}} % \put(53,0){\sx{.6}{$0.5$}} % \put(106,0){\sx{.66}{$1$}} % \put(206,0){\sx{.66}{$2$}} % \put(306,0){\sx{.66}{$3$}} % \put(406,0){\sx{.66}{$4$}} % \put(504,0){\sx{.7}{$x$}} % \put(6,20){\rot{90} \sx{.7}{$f\!=\!0$} \ero} % \put(16,20){\rot{88} \sx{.7}{$f\!=\!0.01$} \ero} % \put(26,19){\rot{82} \sx{.7}{$f\!=\!0.1$} \ero} % \put(46,17){\rot{73} \sx{.7}{$f\!=\!0.2$} \ero} % \put(68,17){\rot{64} \sx{.7}{$f\!=\!0.3$} \ero} % \put(93,17){\rot{55} \sx{.7}{$f\!=\!0.4$} \ero} % \put(123,18){\rot{40} \sx{.7}{$f\!=\!0.5$} \ero} % \put(166,21){\rot{22} \sx{.7}{$f\!=\!0.4$} \ero} % \put(224,20){\rot{13} \sx{.7}{$f\!=\!0.3$} \ero} % \put(314,18){\rot{4} \sx{.7}{$f\!=\!0.2$} \ero} % \put(424,10){\rot{3} \sx{.7}{$f\!=\!0.1$} \ero} % \end{picture} % } % \end{document}

Testing of the implementaitons of Yulya and ArcYulya
For the testing, the following two quantities should be plotted in the $x,a$ coordinates:
 * Yulya(a,ArcYulya2(a,x))-x
 * ArcYulya2(a,Yulya(a,x))-x

Permission to reuse
Copleft 2011 by Dmitrii Kouznetsov. The image and its generators may be used for free, attribute the source.