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.
References
C++ generator of curves
#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;
#include<complex>
typedef std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#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<N1;n++) { Y[n]=1.09*sinh((3./200.)*(n-200)); printf("%3d %9.6f\n",n,Y[n]); }
for(m=-10;m<11;m++){M(.1*m,-1) L(.1*m,1) }
for(n=-10;n<11;n++){M( -1,.1*n) L(1,.1*n)}
fprintf(o,".001 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){ g[m*N1+n]=99999999;
f[m*N1+n]=99999999;}
DO(m,M1){x=X[m]; if(m/10*10==m) printf("x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.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
File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 06:10, 1 December 2018 | | 2,878 × 2,822 (1.92 MB) | Maintenance script (talk | contribs) | Importing image file |
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