Difference between revisions of "File:Sutralomap.jpg"

Latest revision as of 03:06, 10 January 2026

Fig.20.7 from page 280 of book «Superfunctions»^[1], 2020.

The image is used also as Рис.20.7 at page 289 of the Russian version «Суперфункции» ^[2], 2014.

And also as Figure 3 at page 6529 of publication ^[3] at Applied Mathematical Sciences, 2013.

The figure shows the overlappings of complex map of function

\(z \mapsto -\ln(-z)\)

with two its approximations with entire functions,

\(f_2(z)=\mathrm{SuTra}(2z) + \ln(2)\) , left, and

\(f_4(z)=\mathrm{SuTra}(4z) + \ln(4)\) , right.

For function \(f\), levels are shown with

\(u\!+\!\mathrm i v= f(x\!+\!\mathrm i y)\)

in the \(x\), \(y\) plane.

Function SuTra as Entire Function with Logarithmic Asymptotic is described in 2013 ^[3]. It appears as superfunction for the Trappmann function \( \mathrm{tra} = z \mapsto z + \exp(z) \) in role of the transfer function. Before year 2013, such a superfunction was considered to be difficult to construct if at al, as the Trappmann function has no fixed points.

C++ generator of map of z -> -ln(-z)

/* Files ado.cin, conto.cin should be loaded to the working directory in order to compile the code below.*/

#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 "tania.cin"
//#include "LambertW.cin"
//#include "SuZex.cin"
//#include "AuZex.cin" 
//z_type tra(z_type z) {return z+exp(z);}
// z_type sutra(z_type z){ if( Re(z)<2. || fabs(Im(z))>2. ) return log(suzex(z));
//z_type sutra(z_type z){ if( Re(z)<2. ) return log(suzex(z));
//                                       return tra(sutra(z-1.));}

#include"sutran.cin"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
//DB x1=-1.1259817765745026; DO(n,8){ y=Re(suzex(x1)); x=y-1.; x1+=-1.2*x; printf("%18.16f %18.16f\n", x1,y);} getchar();
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("sutranmap.eps","w");  ado(o,2002,2002);
FILE *o;o=fopen("mlogmap.eps","w");  ado(o,2002,2002);
fprintf(o,"1001 1001 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-10+.02*(m-.5);
DO(n,N1) Y[n]=-10+.02*(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(m,-10) L(m,10)  }
for(n=-10;n<11;n++){M(  -10,n) L(10,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=AuZex01(z-1.);
// c=AuZexAsy(LambertW(z))+1.;
//c=log(suzex(z));
//  c=sutran(z);
  c=-log(-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>-19 && p<19 && ( x<2. || (fabs(q)>1.e-12 && fabs(p)>1.e-12)) ){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");
 p=1.2;q=.4;
/*  p=9;q=.16;
 conto(o,g,w,v,X,Y,M,N,(15.3 ),-p,p); fprintf(o,".01 W .4 1 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(15.  ),-p,p); fprintf(o,".02 W 1 0 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14.7 ),-p,p); fprintf(o,".01 W 1 .5 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14.  ),-p,p); fprintf(o,".01 W .2 .2 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,".03 W 0 0 1 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,".03 W 0 1 1 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,".04 W 0 .5 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,".02 W .5 0 0 RGB S\n");
*/
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.  ),-2*p,2*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");
// 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 mlogmap.eps"); 
      system(    "open mlogmap.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

C++ generator of map of f_2

/* Files ado.cin, conto.cin, sutran.cin should be loaded to the working directory in order to compile the code below */

#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 "tania.cin"
//#include "LambertW.cin"
//#include "SuZex.cin"
//#include "AuZex.cin" 
//z_type tra(z_type z) {return z+exp(z);}
// z_type sutra(z_type z){ if( Re(z)<2. || fabs(Im(z))>2. ) return log(suzex(z));
//z_type sutra(z_type z){ if( Re(z)<2. ) return log(suzex(z));
//                                       return tra(sutra(z-1.));}

#include"sutran.cin"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
//DB x1=-1.1259817765745026; DO(n,8){ y=Re(suzex(x1)); x=y-1.; x1+=-1.2*x; printf("%18.16f %18.16f\n", x1,y);} getchar();
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("sutra2map.eps","w");  ado(o,2002,2002);
fprintf(o,"1001 1001 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-10+.02*(m-.5);
DO(n,N1) Y[n]=-10+.02*(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(m,-10) L(m,10)  }
for(n=-10;n<11;n++){M(  -10,n) L(10,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=AuZex01(z-1.);
// c=AuZexAsy(LambertW(z))+1.;
//c=log(suzex(z));
  c=sutran(2.*z)+log(2.);
// 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>-19 && p<19 && ( x<2. || (fabs(q)>1.e-12 && fabs(p)>1.e-12)) ){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");
 p=1.2;q=.4;
/*  p=9;q=.16;
 conto(o,g,w,v,X,Y,M,N,(15.3 ),-p,p); fprintf(o,".01 W .4 1 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(15.  ),-p,p); fprintf(o,".02 W 1 0 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14.7 ),-p,p); fprintf(o,".01 W 1 .5 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14.  ),-p,p); fprintf(o,".01 W .2 .2 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,".03 W 0 0 1 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,".03 W 0 1 1 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,".04 W 0 .5 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,".02 W .5 0 0 RGB S\n");
*/
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.  ),-2*p,2*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");
// 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 sutra2map.eps"); 
      system(    "open sutra2map.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

C++ generator of map of f_4

/* Files ado.cin, conto.cin, sutran.cin should be loaded to the working directory in order to compile the code below*/

#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 "tania.cin"
//#include "LambertW.cin"
//#include "SuZex.cin"
//#include "AuZex.cin" 
//z_type tra(z_type z) {return z+exp(z);}
// z_type sutra(z_type z){ if( Re(z)<2. || fabs(Im(z))>2. ) return log(suzex(z));
//z_type sutra(z_type z){ if( Re(z)<2. ) return log(suzex(z));
//                                       return tra(sutra(z-1.));}

#include"sutran.cin"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
//DB x1=-1.1259817765745026; DO(n,8){ y=Re(suzex(x1)); x=y-1.; x1+=-1.2*x; printf("%18.16f %18.16f\n", x1,y);} getchar();
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("sutra4map.eps","w");  ado(o,2002,2002);
fprintf(o,"1001 1001 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-10+.02*(m-.5);
DO(n,N1) Y[n]=-10+.02*(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(m,-10) L(m,10)  }
for(n=-10;n<11;n++){M(  -10,n) L(10,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=AuZex01(z-1.);
// c=AuZexAsy(LambertW(z))+1.;
//c=log(suzex(z));
  c=sutran(4.*z)+log(4.);
// 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>-19 && p<19 && ( x<2. || (fabs(q)>1.e-12 && fabs(p)>1.e-12)) ){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");
 p=1.2;q=.4;
/*  p=9;q=.16;
 conto(o,g,w,v,X,Y,M,N,(15.3 ),-p,p); fprintf(o,".01 W .4 1 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(15.  ),-p,p); fprintf(o,".02 W 1 0 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14.7 ),-p,p); fprintf(o,".01 W 1 .5 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14.  ),-p,p); fprintf(o,".01 W .2 .2 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,".03 W 0 0 1 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,".03 W 0 1 1 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,".04 W 0 .5 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,".02 W .5 0 0 RGB S\n");
*/
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.  ),-2*p,2*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");
// 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 sutra4map.eps"); 
      system(    "open sutra4map.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex combiner

%

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\usepackage{graphics}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\begin{document}
\parindent 0pt
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%\zoomax % 
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%

References

↑ https://mizugadro.mydns.jp/BOOK/468.pdf D.Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
↑ https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
https://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014
↑ ^3.0 ^3.1 http://www.m-hikari.com/ams/ams-2013/ams-129-132-2013/kouznetsovAMS129-132-2013.pdf D.Kouznetsov. Entire Function with Logarithmic Asymptotic. Applied Mathematical Sciences, Vol. 7, 2013, no. 131, 6527 - 6541

Keywords

«Entire Function with Logarithmic Asymptotic», «Logarithm», «Superfunction», «Superfunctions», «SuTra», «Trappmann function», «[[]]»,

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[1] ttps://mizugadro.mydns.jp/BOOK/468.pdf D.Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.

[2] ttps://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
https://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014

[h-3] 3.0 ^3.1 http://www.m-hikari.com/ams/ams-2013/ams-129-132-2013/kouznetsovAMS129-132-2013.pdf D.Kouznetsov. Entire Function with Logarithmic Asymptotic. Applied Mathematical Sciences, Vol. 7, 2013, no. 131, 6527 - 6541

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
+{{oq|Sutralomap.jpg|Original file ‎(2,533 × 1,254 pixels, file size: 1.26 MB, MIME type: image/jpeg)|600}}
-Overlappings of [[complex map]] of function
+Fig.20.7 from page 280 of book «[[Superfunctions]]»<ref>
-$z \mapsto -\ln(-z)$
+https://mizugadro.mydns.jp/BOOK/468.pdf
+D.Kouznetsov. [[Superfunctions]].  [[Lambert Academic Publishing]], 2020.
+</ref>, 2020.
+The image is used also as Рис.20.7 at page 289 of the Russian version «[[Суперфункции]]»
-with two its approximations with entire functions,
+<ref>
+https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
+https://mizugadro.mydns.jp/BOOK/202.pdf
+Д.Кузнецов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014
+</ref>, 2014.
+And also as Figure 3 at page 6529 of publication <ref name="h"/> at [[Applied Mathematical Sciences]], 2013.
-$f_2(z)=\mathrm{SuTra}(2z) + \ln(2)$ , left, and
+The figure shows the overlappings of [[complex map]] of function
-$f_4(z)=\mathrm{SuTra}(4z) + \ln(4)$ , right.
+\(z \mapsto -\ln(-z)\)
-for function $f$, levels are shown with
+with two its approximations with entire functions,
-$u\!+\!\mathrm i v= f(x\!+\!\mathrm i y)$
+\(f_2(z)=\mathrm{SuTra}(2z) + \ln(2)\) , left, and
-in the $x$, $y$ plane.
+\(f_4(z)=\mathrm{SuTra}(4z) + \ln(4)\) , right.
-This image is used as figure 20.7 of the Book
-[[Superfunction]]  (In Russian, 2014; the English version is in preparation, 2015)
-<ref>
-https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
-http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf <br>
-http://mizugadro.mydns.jp/BOOK/202.pdf
-Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014
-</ref>.
+For function \(f\), levels are shown with
-Function [[SuTra]]
-as entire function with logarithmic asymptotic is described also in year 2013 at Hikari
-<ref>
- http://www.m-hikari.com/ams/ams-2013/ams-129-132-2013/kouznetsovAMS129-132-2013.pdf D.Kouznetsov. Entire Function with Logarithmic Asymptotic. Applied Mathematical Sciences, Vol. 7, 2013, no. 131, 6527 - 6541
-</ref>.
+\(u\!+\!\mathrm i v= f(x\!+\!\mathrm i y)\)
-==References==
-<references/>
+in the \(x\), \(y\) plane.
+Function [[SuTra]]
-==[[C++]] generator of map of $z\mapsto -\ln(-z)$ ==
+as [[Entire Function with Logarithmic Asymptotic]] is described in 2013 <ref name="h">
+http://www.m-hikari.com/ams/ams-2013/ams-129-132-2013/kouznetsovAMS129-132-2013.pdf D.Kouznetsov. [[Entire Function with Logarithmic Asymptotic]]. [[Applied Mathematical Sciences]], Vol. 7, 2013, no. 131, 6527 - 6541
+</ref>. It appears as [[superfunction]] for the [[Trappmann function]]
+\( \mathrm{tra} = z \mapsto z + \exp(z) \)
+in role of the [[transfer function]]. Before year 2013, such a superfunction was considered to be
+difficult to construct if at al, as the [[Trappmann function]] has no [[fixed point]]s.
+==[[C++]] generator of map of z -> -ln(-z) ==
-Files [[ado.cin]], [[conto.cin]]
+/* Files [[ado.cin]], [[conto.cin]]
-should be loaded to the working directory in order to compile the code below
+should be loaded to the working directory in order to compile the code below.*/
-<poem><nomathjax><nowiki>
+<pre>
 #include <math.h>
 #include <stdio.h>
@@ Line 132: / Line 136: @@
       getchar(); system("killall Preview"); // For macintosh
 }
+</pre>
-</nowiki></nomathjax></poem>
-==[[C++]] generator of map of $f_2$ ==
+==[[C++]] generator of map of f_2 ==
+/* Files
-Files [[ado.cin]],
+[[ado.cin]],
 [[conto.cin]],
 [[sutran.cin]]
-should be loaded to the working directory in order to compile the code below
+should be loaded to the working directory in order to compile the code below */
+<pre>
-<poem><nomathjax><nowiki>
 #include <math.h>
 #include <stdio.h>
@@ Line 232: / Line 236: @@
       getchar(); system("killall Preview"); // For macintosh
 }
+</pre>
-</nowiki></nomathjax></poem>
-==[[C++]] generator of map of $f_4$==
+==[[C++]] generator of map of f_4==
+/*
 Files [[ado.cin]],
 [[conto.cin]],
 [[sutran.cin]]
-should be loaded to the working directory in order to compile the code below
+should be loaded to the working directory in order to compile the code below*/
+<pre>
-<poem><nomathjax><nowiki>
 #include <math.h>
 #include <stdio.h>
@@ Line 332: / Line 336: @@
       getchar(); system("killall Preview"); // For macintosh
 }
+</pre>
-</nowiki></nomathjax></poem>
 ==[[Latex]] combiner==
+%<pre>
-<poem><nomathjax><nowiki>
 \documentclass[12pt]{article}
 \paperwidth 1216px
@@ Line 432: / Line 434: @@
 \end{picture}}
 \end{document}
+%</pre>
+==References==
+{{ref}}
+{{fer}}
+==Keywords==
+«[[Entire Function with Logarithmic Asymptotic]]»,
-</nowiki></nomathjax></poem>
+«[[Logarithm]]»,
+«[[Superfunction]]»,
+«[[Superfunctions]]»,
+«[[SuTra]]»,
+«[[Trappmann function]]»,
+«[[]]»,
 [[Category:Book]]
@@ Line 442: / Line 456: @@
 [[Category:Latex]]
 [[Category:Superfunction]]
+[[Category:Superfunctions]]
 [[Category:SuTra]]
 [[Category:Trappmann function]]