File:SuZexD1mapT.png

Complex map of function SuZex, which is entire superfunction of Zex, $\mathrm{zex}(z)=z\exp(z)$.

$u\!+\!\mathrm i v = \mathrm{SuZex}(c\!+\!\mathrm i y)$

SuZex is built-up at the fixed point zero from asymptotic behavior; the approximation below is implemented:

$\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ \displaystyle

f(z)= \frac{-1}{z} +\frac{ \ln(-z)}{2 z^2}+ \frac{-.05\ln(-z)^2-.02\ln(-z)-.4}{z^3}$ For integer $n$,

$\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \displaystyle

F_n(z)=\mathrm{zex}^n\Big( z\!-\!n) \Big)$ The constant $x_n$ is chosen as solution of equation $F_n(x_n)\!=\!1$. Then, the superfunction is evaluated as iteration

$\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \displaystyle

\mathrm{SuZex}(z)=F_n(x_n\!+\!z)$ for integer $n$.

The generator below uses value $n\!=\!16$, which is sufficient to get the camera-ready copy. For the precise computation, more terms in the expansion (1) should be calculated. The precise implementation (with 14 decimal digits) is loaded as SuZex.cin.

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 complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
// #include "fsexp.cin"
//#include "fslog.cin"

z_type zex(z_type z){ return z*exp(z);} 

int Nitera=16;

z_type suzex0b(z_type z){ return -1./z; }

// z_type suzex0(z_type z){ z_type L=log(-z); return (-1.+.5*L/z)/z; }

z_type suzex0(z_type z){ z_type L=log(-z); return (-1.+(.5*L + (-.05*L*L-.02*L-.4)/z)/z)/z; }

z_type suzexn(int n, z_type z){int m;  z-=0.+n; z=suzex0(z); DO(m,n) z=zex(z); return z; } 

main(){ int j,k,m,n; DB x1,x,y, p,q, t; z_type z,c,d, cu,cd;
x1=-1.04;
DO(n,18){ y=Re(suzexn(Nitera,x1)); x=y-1.; x1-=1.5*x; printf("%18.16f %18.16f\n", x1,y);}
getchar();

int M=601,M1=M+1;
int N=401,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("suZexMap.eps","w");  ado(o,1202,1202);
fprintf(o,"601 601 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-6.+.02*(m-.1);
//DO(n,N1) Y[n]=-5.+.02*(n-.5); 
for(n=0;n<N1;n++) { Y[n]=0.6*sinh((3./200.)*(n-200.5)); printf("%3d %9.6f\n",n,Y[n]); }
for(m=-6;m<7;m++) {M(m,-6)L(m,6)}
for(n=-6;n<7;n++) {M(  -6,n)L(6,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){      g[m*N1+n]=999;
                       f[m*N1+n]=999;}
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=suzexn(Nitera,z+x1);
        p=Re(c); q=Im(c);
        if(p>-19 && p<19 &&  fabs(q)>1.e-12 && fabs(p)>1.e-12) g[m*N1+n]=p;
        if(p>-19 && p<19 &&  fabs(q)>1.e-12 && fabs(p)>1.e-12) f[m*N1+n]=q;
      }
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");
p=2.;q=.3;

for(m=-8;m<8;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".007 W 0 .6 0 RGB S\n");
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".007 W .9 0 0 RGB S\n");
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".007 W 0 0 .9 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".02 W .8 0 0 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".02 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".02 W .5 0 .5 RGB S\n");
for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");

//#include "plofu.cin"
fprintf(o,"0 setlinejoin 0 setlinecap\n");
fprintf(o,"showpage\n");
fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o);  free(f); free(g); free(w);
      system("epstopdf SuZexMap.eps"); 
      system(    "open SuZexMap.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex generator of labels

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