File:Arcnembrat.jpg

Complex map of function ArcNemBra,

$u\!+\!\mathrm i v=\mathrm{ArcNemBra}(x\!+\!\mathrm i y)$

Function NemBra expresses position of the branchpoint of the Nemtsov function

$\mathrm{Nem}_q(z)=z+z^3+qz^4$

If $z=\mathrm{NemBra}(q)$, then $\mathrm{Nem}_q'(z)=0$

ArcNemBra is elementary function. For moderate values of argument, for the evaluation the explicit representation is used,

$\displaystyle \mathrm{ArcNemBra}(z) = - \frac{1\!+\!3z^2}{4\, z^3}$

For very small or very large values of $|z|$, the asymptotic expansions can be used.

C++ generator of curves
//using namespace std; typedef std::complex z_type;
 * 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 "conto.cin"

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; int M=1001,M1=M+1; int N=1001,N1=N+1; DB X[M1],Y[N1]; DB *g, *f, *w; // w is working array. g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB))); f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB))); w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB))); char v[M1*N1]; // v is working array FILE *o;o=fopen("arcnembra.eps","w"); ado(o,2002,2002); fprintf(o,"1001 1001 translate\n 1000 1000 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); DO(m,M1) X[m]=-1+.002*(m-.5); DO(n,N1) Y[n]=-1+.002*(n-.5); //for(n=0;n.019)

// c=-1./z; c=-(1.+3.*z*z)/(4.*z*z*z);

// p=abs(c-z)/(abs(c)+abs(z)); p=-log(p)/log(10.); if(p>0 && p<17) g[m*N1+n]=p; p=Re(c); q=Im(c); if(p>-100000 && p<100000 && q>-100000 && q<100000 ){ g[m*N1+n]=p;f[m*N1+n]=q;} }} fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=10.;q=0.4;

for(m=-4;m<4;m++)for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".001 W 0 .6 0 RGB S\n"); for(m=0;m<4;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".001 W .8 0 0 RGB S\n"); for(m=0;m<4;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".001 W 0 0 .8 RGB S\n"); for(m= 1;m<6;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".004 W .8 0 0 RGB S\n"); for(m= 1;m<6;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".004 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-1000000,1000000); fprintf(o,".01 W 1 0 1 RGB S\n"); for(m=-6;m<7;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".004 W 0 0 0 RGB S\n"); // fprintf(o,"0 setlinejoin 0 setlinecap\n"); // M(-10,0)L(0,0) fprintf(o,"1 1 1 RGB .02 W S\n"); //#include "plofu.cin" fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); free(f); free(g); free(w); system("epstopdf arcnembra.eps"); system( "open arcnembra.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} %\paperwidth 174pt %\paperheight 620pt \paperwidth 2080pt \paperheight 2040pt \topmargin -96pt %\oddsidemargin -72pt \oddsidemargin -22pt % Warning: labels will be out of range of the figure! \textheight 3000pt \textwidth 3000pt \parindent 0pt \usepackage{graphicx} \usepackage{rotating} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \JP[1] – \begin{document} \begin{picture}(2042,2012) %\put(1020,1014){\ing{nembraplo}} \put(20,14){\ing{arcnembra}} %\put(20,14){\ing{arcnembra1}} %small file \put(-22,2006){\sx{3.1}{$y$}} \put(-34,1908){\sx{3}{$0.9$}} \put(-34,1808){\sx{3}{$0.8$}} \put(-34,1708){\sx{3}{$0.7$}} \put(-34,1608){\sx{3}{$0.6$}} \put(-34,1508){\sx{3}{$0.5$}} \put(-34,1408){\sx{3}{$0.4$}} \put(-34,1308){\sx{3}{$0.3$}} \put(-34,1208){\sx{3}{$0.2$}} \put(-34,1108){\sx{3}{$0.1$}} \put(-24,1008){\sx{3}{$0$}} \put(-54,908){\sx{3}{$-0.1$}} \put(-54,808){\sx{3}{$-0.2$}} \put(-54,708){\sx{3}{$-0.3$}} \put(-54,608){\sx{3}{$-0.4$}} \put(-54,508){\sx{3}{$-0.5$}} \put(-54,408){\sx{3}{$-0.6$}} \put(-54,308){\sx{3}{$-0.7$}} \put(-54,208){\sx{3}{$-0.8$}} \put(-54,108){\sx{3}{$-0.9$}} \put(-44, 8){\sx{3}{$-1$}} \put(-2, -12){\sx{3}{$-1$}} \put(88, -12){\sx{3}{$-0.9$}} \put(188, -12){\sx{3}{$-0.8$}} \put(288, -12){\sx{3}{$-0.7$}} \put(388, -12){\sx{3}{$-0.6$}} \put(488, -12){\sx{3}{$-0.5$}} \put(588, -12){\sx{3}{$-0.4$}} \put(688, -12){\sx{3}{$-0.3$}} \put(788, -12){\sx{3}{$-0.2$}} \put(888, -12){\sx{3}{$-0.1$}} \put(1017, -12){\sx{3}{$0$}} \put(1106, -12){\sx{3}{$0.1$}} \put(1206, -12){\sx{3}{$0.2$}} \put(1306, -12){\sx{3}{$0.3$}} \put(1406, -12){\sx{3}{$0.4$}} \put(1506, -12){\sx{3}{$0.5$}} \put(1606, -12){\sx{3}{$0.6$}} \put(1706, -12){\sx{3}{$0.7$}} \put(1806, -12){\sx{3}{$0.8$}} \put(1906, -12){\sx{3}{$0.9$}} \put(2009, -12){\sx{3.1}{$x$}} \put(330,1312){\rot{-68}\sx{5}{$v\!=\!1$}\ero}		\put(1660,1180){\rot{63}\sx{5}{$v\!=\!1$}\ero} \put(48,1128){\rot{-18}\sx{5}{$v\!=\!0.2$}\ero}	\put(1840,1074){\rot{17}\sx{5}{$v\!=\!0.2$}\ero} \put(40,1004){\sx{5}{$v\!=\!0$}}		\put(1840,1004){\sx{5}{$v\!=\!0$}} \put(58,884){\rot{16}\sx{5}{$v\!=\!-0.2$}\ero}	\put(1790,950){\rot{-20}\sx{5}{$v\!=\!-0.2$}\ero} \put(340,552){\rot{-50}\sx{5}{$v\!=\!-1$}\ero}		\put(1660,480){\rot{63}\sx{5}{$v\!=\!-1$}\ero} % \put(92,1210){\rot{42}\sx{5}{$u\!=\!1$}\ero} \put(368,1040){\rot{64}\sx{5}{$u\!=\!2$}\ero} \put(1850,700){\rot{40}\sx{5}{$u\!=\!-1$}\ero} % \put(1040,1800){\rot{90}\sx{5}{$u\!=\!0$}\ero} \put(780,1934){\rot{25}\sx{5}{$u\!=\!0$}\ero}% \put(1168,1986){\rot{-28}\sx{5}{$u\!=\!0$}\ero}% \put(1066,426){\rot{37}\sx{5}{$v\!=\!0$}\ero} \put(1066,32){\rot{57}\sx{5}{$v\!=\!-0.5$}\ero} % \put(1040,130){\rot{90}\sx{5}{$u\!=\!0$}\ero} \put(1168,20){\rot{26}\sx{5}{$u\!=\!0$}\ero} % \put(902,1826){\rot{57}\sx{5}{$v\!=\!0.5$}\ero} \put(1060,1966){\rot{-58}\sx{5}{$v\!=\!0.5$}\ero} \put(910,1520){\rot{37}\sx{5}{$v\!=\!0$}\ero} \put(960,870){\rot{60}\sx{5}{$v\!=\!0$}\ero} %

\end{picture} \end{document} \JP{ \put(1022.3,1016.3){\ing{nembraplo}} %\put(20,14){\ing{arcnembra}} \put(1008,1614){\sx{1.4}{$y$}} \put(1000,1514){\sx{1.4}{$0.5$}} \put(1000,1414){\sx{1.4}{$0.4$}} \put(1000,1314){\sx{1.4}{$0.3$}} \put(1000,1214){\sx{1.4}{$0.2$}} \put(1000,1114){\sx{1.4}{$0.1$}} \put(1008,1014){\sx{1.4}{$0$}} \put(1021,1001){\sx{1.4}{$0$}} \put(1060,1001){\sx{1.4}{$0.05$}} \put(1112,1001){\sx{1.4}{$0.1$}} \put(1165,1002){\sx{1.4}{$x$}} %%\put(18,601){\sx{1.1}{$q\!=\!0$}} \put(1002,1583){\sx{1.1}{$q\!=\!0$}} %%\put(63,593){\sx{1.1}{$q\!=\!0.2$}} \put(1030,1576){\sx{1.1}{$q\!=\!0.2$}} %%\put(97,574){\sx{1.1}{$q\!=\!0.4$}} \put(1062,1558){\sx{1.1}{$q\!=\!0.4$}} %%\put(122,550){\sx{1.1}{$q\!=\!0.6$}} \put(1085,1535){\sx{1.1}{$q\!=\!0.6$}} %%\put(137,528){\sx{1.1}{$q\!=\!0.8$}} \put(1100,1514){\sx{1.1}{$q\!=\!0.8$}} %%\put(148,506){\sx{1.1}{$q\!=\!1$}} \put(1121,1494){\sx{1.1}{$q\!=\!1$}} \put(1137,1424){\sx{1.1}{$q\!=\!2$}} \put(1138,1380){\sx{1.1}{$q\!=\!3$}} \put(1136,1350){\sx{1.1}{$q\!=\!4$}} \put(1133,1330){\sx{1.1}{$q\!=\!5$}} \put(1129,1310){\sx{1.1}{$q\!=\!6$}} \put(1122,1286){\sx{1.1}{$q\!=\!8$}} \put(1110,1268){\sx{1.1}{$q\!=\!10$}} \put(1090,1216){\sx{1.1}{$q\!=\!20$}} \put(1082,1192){\sx{1.1}{$q\!=\!30$}} \put(1088,1127){\sx{1.1}{$q\!=\!100$}} \put(1077,1104){\sx{1.1}{$q\!=\!200$}} \put(1054,1064){\sx{1.1}{$q\!=\!1000$}} \put(1039,1036){\sx{1.1}{$q\!=\!10000$}} }%JP