File:Arcnemap11.jpg

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Complex map of function ArcNem, which is inverse function of the Nemtsov function Nem. Values of parameters: $p\!=\!1$, $q\!=\!1$.

$x\!+\!\mathrm i y = \mathrm{ArcNem}_{1,1}(x\!+\mathrm i y)$

The branch points are marked with additional grid lines.

The cut lines are horizontal, along $y\!=\!0$ and $y\!=\!\pm \Im(\mathrm{Ni}(p,q))$

where $Ni(p,q)$ are complex soludions of equation $\mathrm{Nem}_{p,q}'=0$. The Nemtsov function is 4th order polynomial of special king:

$\mathrm{Nem}_{p,q}(z)=z+pz^3+qz^4$

For the Nemtsov function, none of the algorithms of construction of superfuncitons reported in the preliminary version of book Superfunctions can be applied "as is", even the algorithm for the superfunction of sin [1].

For this reason, this function needs the special name to handle.

Description

Roots of the four–order polynomial can be expressed through elementary function, as some combination of the square roots and cubic roots. However, the careful handling of the cut lines is required to avoid confusions with the non–physical solutions.

Another inverse funciton is called ArkNem; it has two vertical cut lines.

References

  1. ↑ http://mizugadro.mydns.jp/PAPERS/2014susinL.pdf
    http://mizugadro.mydns.jp/PAPERS/2014susin.pdf
    http://www.pphmj.com/references/8246.htm D.Kouznetsov. Super sin. Far East Jourmal of Mathematical Science, v.85, No.2, 2014, pages 219-238.

C++ generator of the map


#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"

#include "arcnem.cin"

int main(){ int Max; int j,k,m,n; DB x,y, p,q,P,Q, t, sr,si,tr,ti; z_type z,c,d,s;
P=1; // WARNING: here p,q are already in use in two different meanings for graphics.
Q=1; // WARNING: in this routine, the capital letters are used to denote coefficents
s=nemsingu(P,Q); //WARNING: it assign valies to global variables DB NEMTSOV@
tr=NEMTSOVr;
ti=NEMTSOVi;

printf("P,Q : %6.2lf RI: %6.2lf %20.16lf %20.16lf\n", P, Q,NEMTSOVR,NEMTSOVI);
printf("GLOBAL: %6.2lf ri: %6.2lf %20.16lf %20.16lf\n",NEMTSOVp,NEMTSOVq,NEMTSOVr,NEMTSOVi);

int M=401,M1=M+1;
int N=401,N1=N+1;
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
char v[M1*N1]; // v is working array
//FILE *o;o=fopen("nemp10q10m4.eps","w");ado(o,402,402);
FILE *o;o=fopen("arcnep10q10m4.eps","w");ado(o,402,402);
fprintf(o,"201 201 translate\n 100 100 scale\n");
DO(m,M1) X[m]=-2.+.01*(m-.5);
DO(n,N1) Y[n]=-2.+.01*(n-.5);
for(m=-2;m<4;m++){ M(m,-2)L(m,2)}
for(n=-2;n<4;n++){ M(-2,n)L(2,n)}

M(tr, -2) L(tr, 2)
M(-2, ti) L(2, ti)
M(-2,-ti) L(2,-ti)
fprintf(o,".003 W 0 0 0 RGB S\n");

DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);

//c=arcnem0(P,Q,z);
c=arcnempq(z);
//c=nem(P,Q,c);
//p=abs(c-z)/(abs(c)+abs(z)); p=-log(p)/log(10.); if(p>-85 && p<85) g[m*N1+n]=p;
p=Re(c); q=Im(c);
        if(p>-40 && p<40 &&
          q >-40 && q<40 ) { g[m*N1+n]=p; f[m*N1+n]=q; }
        }}

fprintf(o,"1 setlinejoin 1 setlinecap\n");
//p=40.;q=.5;
//#include"plofu.cin"
p=.5;q=1;

for(m=-9;m<9;m++)for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".002 W 0 .6 0 RGB S\n");
for(m=0;m<19;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);
                                                fprintf(o,".002 W .9 0 0 RGB S\n");
for(m=0;m<19;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".002 W 0 0 .9 RGB S\n");

for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".005 W .8 0 0 RGB S\n");
for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".005 W 0 0 .8 RGB S\n");
                   conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".005 W .5 0 .5 RGB S\n");
for(m=-21;m<22;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".005 W 0 0 0 RGB S\n");

/*
conto(o,g,w,v,X,Y,M,N,15.4,-p,p);fprintf(o,".02 W 1 0 1 RGB S\n");
conto(o,g,w,v,X,Y,M,N,15.,-p,p);fprintf(o,".02 W 0 0 1 RGB S\n");
conto(o,g,w,v,X,Y,M,N,14.,-p,p);fprintf(o,".02 W 0 1 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,13.,-p,p);fprintf(o,".02 W 1 0 0 RGB S\n");
*/
/*
conto(o,g,w,v,X,Y,M,N,15.5,-p,p);fprintf(o,".02 W .3 0 .3 RGB S\n");
conto(o,g,w,v,X,Y,M,N,15.,-p,p);fprintf(o,".02 W .8 0 .8 RGB S\n");
conto(o,g,w,v,X,Y,M,N,14.,-p,p);fprintf(o,".01 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,13.,-p,p);fprintf(o,".01 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,12.,-p,p);fprintf(o,".02 W 0 0 .8 RGB S\n");
conto(o,g,w,v,X,Y,M,N,11.,-p,p);fprintf(o,".01 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,10.,-p,p);fprintf(o,".01 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,9.,-p,p);fprintf(o,".02 W 0 .6 .8 RGB S\n");
conto(o,g,w,v,X,Y,M,N,8.,-p,p);fprintf(o,".01 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,7.,-p,p);fprintf(o,".01 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,6.,-p,p);fprintf(o,".02 W 0 .7 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,5.,-p,p);fprintf(o,".01 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,4.,-p,p);fprintf(o,".01 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,3.,-p,p);fprintf(o,".02 W 1 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,2.,-p,p);fprintf(o,".01 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,1.,-p,p);fprintf(o,".03 W .5 0 0 RGB S\n");
*/

/*
fprintf(o,"0 setlinecap\n");
M(M_E,0)L(-10,0) fprintf(o,".08 W 1 1 1 RGB S\n");
DO(m,36){M(M_E-.4*(m),0)L(M_E-.4*(m+.5),0)} fprintf(o,".09 W 1 0 1 RGB S\n");
*/

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
        system("epstopdf arcnep10q10m4.eps");
        system( "open arcnep10q10m4.pdf"); //mac

getchar(); system("killall Preview");// mac
}

Latex generator of labels


 \documentclass[12pt]{article}
 \usepackage{graphics}
 \paperwidth 430pt
 \paperheight 426pt
 \usepackage{geometry}
 \usepackage{rotating}
 \textwidth 1260pt
 \textheight 1260pt
 \topmargin -108pt
 \oddsidemargin -72pt
 \parindent 0pt
 \pagestyle{empty}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}
\begin{picture}(424,424)
%\put(24,20){\ing{arknemmap1010}}
\put(24,20){\ing{arcnep10q10m4}}
%\put(50,40){\ing{sunem10q10ma6a}}
%\put(50,40){\ing{sunem10q10ma6s}}
\put(11,415){\sx{1.8}{$y$}}
\put(10,314){\sx{1.8}{$1$}}
\put(10,214){\sx{1.8}{$0$}}
\put(-5,114){\sx{1.8}{$-1$}}
\put(-5, 14){\sx{1.8}{$-2$}}
\put( 5, 1){\sx{1.8}{$-2$}}
\put(105, 1){\sx{1.8}{$-1$}}
\put(221, 1){\sx{1.8}{$0$}}
\put(321, 1){\sx{1.8}{$1$}}
\put(418,1){\sx{1.8}{$x$}}
%\put(32,154){\sx{1.9}{\rot{40} $u\!=\!-1$\ero}}
%\put(58,116){\sx{1.8}{\rot{57} $u\!=\!-0.9$\ero}}
%\put(94,100){\sx{1.8}{\rot{70} $u\!=\!-0.8$\ero}}
%\put(130,100){\sx{1.8}{\rot{80} $u\!=\!-0.7$\ero}}
%
%\put(159,260){\sx{1.8}{\rot{87} $u\!=\!-0.6$\ero}}
%\put(172,258){\sx{1.8}{\rot{74} $u\!=\!-0.5$\ero}}
%\put(186,256){\sx{1.8}{\rot{65} $u\!=\!-0.4$\ero}}
%
%\put(211,190){\sx{1.8}{\rot{86} $u\!=\!-0.2$\ero}}
\put(230,196){\sx{1.8}{\rot{86} $u\!=\!0$\ero}}
\put(250,188){\sx{1.8}{\rot{88} $u\!=\!0.2$\ero}}
\put(276,188){\sx{1.8}{\rot{86} $u\!=\!0.4$\ero}}
\put(298,188){\sx{1.8}{\rot{86} $u\!=\!0.5$\ero}}
\put(324,188){\sx{1.8}{\rot{86} $u\!=\!0.6$\ero}}
\put(358,188){\sx{1.8}{\rot{86} $u\!=\!0.7$\ero}}
\put(402,188){\sx{1.8}{\rot{86} $u\!=\!0.8$\ero}}
%
%\put(60,398){\sx{1.8}{\rot{3} $v\!=\!0.8$\ero}}
%\put(60,350){\sx{1.8}{\rot{6} $v\!=\!0.7$\ero}}
%\put(68,308){\sx{1.8}{\rot{12} $v\!=\!0.6$\ero}}
%\put(88,280){\sx{1.8}{\rot{18} $v\!=\!0.5$\ero}}
%
\put(190,360){\sx{1.8}{\rot{89} $v\!=\!0.6$\ero}}
\put(238,360){\sx{1.8}{\rot{75} $v\!=\!0.5$\ero}}
%
\put(288,364){\sx{1.8}{\rot{56} $v\!=\!0.4$\ero}}
\put(324,334){\sx{1.8}{\rot{43} $v\!=\!0.3$\ero}}
\put(350,298){\sx{1.8}{\rot{30} $v\!=\!0.2$\ero}}
\put(362,258){\sx{1.8}{\rot{15} $v\!=\!0.1$\ero}}
\put(350,178){\sx{1.8}{\rot{-16} $v\!=\!-0.1$\ero}}
\put(334,140){\sx{1.8}{\rot{-29} $v\!=\!-0.2$\ero}}
\put(306,112){\sx{1.8}{\rot{-46} $v\!=\!-0.3$\ero}}
\put(272,86){\sx{1.8}{\rot{-60} $v\!=\!-0.4$\ero}}
%
%\put(46,89){\sx{1.8}{\rot{-11} $v\!=\!-0.7$\ero}}
%\put(44,36){\sx{1.8}{\rot{-2} $v\!=\!-0.8$\ero}}

\put(148,216){\sx{1.8}{\rot{0.} $v\!=\!0$ \ero}}
\put(26,250){\sx{1.8}{\rot{0.} \bf cut \ero}}
\put(26,217){\sx{1.8}{\rot{0.} \bf cut \ero}}
\put(26,183){\sx{1.8}{\rot{0.} \bf cut \ero}}
%\put(238,280){\sx{1.8}{\rot{90} \bf cut \ero}}
%\put(238,140){\sx{1.8}{\rot{90} \bf cut \ero}}

\put(64,245){\sx{1.8}{\rot{-31} $u\!=\!1$\ero}}
\put(66,187){\sx{1.8}{\rot{31} $u\!=\!1$\ero}}

\put(118,380){\sx{1.8}{\rot{-69} $v\!=\!0.7$\ero}}%
\put(54,358){\sx{1.8}{\rot{-62} $v\!=\!0.8$\ero}}
\put(54,62){\sx{1.8}{\rot{62} $v\!=\!-0.8$\ero}}
\put(124,50){\sx{1.8}{\rot{69} $v\!=\!-0.7$\ero}}
\put(188,34){\sx{1.8}{\rot{86} $v\!=\!-0.6$\ero}}
\end{picture}
\end{document}

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