File:QexpMapT400.jpg

Complex map of function $\sqrt(\exp)= \exp^{1/2}$, Halfiteration of exponential to base $\mathrm e$.

$f=\exp^{1/2}(x+\rm i y$

is shown with lines $u=\Re(f)=\mathrm const$ and lines $v=\Im(f)=\mathrm const$.

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 "fac.cin"
//#include "sinc.cin"
//#include "facp.cin"
//#include "afacc.cin"
//#include "superfac.cin"
#include "superex.cin"
#include "superlo.cin"
DB xL=0.31813150520476413;
DB yL=1.3372357014306895;

#include "conto.cin"

int main(){ int j,k,m,n,n1; DB x,y, p,q, t; z_type z,c,d;
 int M=160,M1=M+1;
 int N=165,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("fig4b.eps","w");ado(o,402,402);
FILE *o;o=fopen("qexpMap.eps","w");ado(o,402,402);
fprintf(o,"201 201 translate\n 20 20 scale\n");
DO(m,M1) X[m]=-8.+.1*m;
DO(n,N1){   y=-8.+.1*n; if(y< -yL) Y[n]=y; else break;}
        Y[n]=-yL-.001; n++;
        Y[n]=-yL+.001; n++;
for(j=n;j<N1;j++){y=-8.+.1*(j-2); if(y<-.02) Y[j]=y; else break;}
        Y[j]= -.02; j++;
        Y[j]= +.02; j++;
for(k=j;k<N1;k++){y=-8.+.1*(k-3); if(y<yL) Y[k]=y; else break;}
        Y[k]= yL-.001; k++;
        Y[k]= yL+.001; k++;
for(n=k;n<N1;n++){y=-8.+.1*(n-5); Y[n]=y;}
for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}}
for(n=-8;n<9;n++){      M(  -8,n)L(8,n)}
fprintf(o,".008 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("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y); 
//      c=afacc(z);
//      c=fac(c);
//      c=arcsuperfac(z);
//      c=superfac(.5+c);
        c=FSLOG(z);
        c=FSEXP(.5+c);
//      d=z;
//      p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.;
        p=Re(c);q=Im(c);        
        if(p>-999 && p<999)     g[m*N1+n]=p;
        if(q>-999 && q<999 && fabs(q)> 1.e-14) f[m*N1+n]=q;
                        }}
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=6.;q=6.;
//#include"plofu.cin"
//p=2;q=1;
for(m=-4;m<4;m++) for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M-20,N,(m+.1*n),-q, q); fprintf(o,".02 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-20,N,-(m+.1*n),-q, q); fprintf(o,".02 W .9 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-20,N, (m+.1*n),-q, q); fprintf(o,".02 W 0 0 .9 RGB S\n");
for(m=1;m<9;m++)  conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".05 W .9 0 0 RGB S\n");
for(m=1;m<9;m++)  conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 .9 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".05 W .6 0 .6 RGB S\n");
for(m=-8;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 0 RGB S\n");
fprintf(o,"0 setlinejoin 0 setlinecap\n");
M(xL, yL)L(-8.1, yL) 
M(xL,-yL)L(-8.1,-yL) fprintf(o,"                  .06 W 1 1 1 RGB S\n");
DO(m,21){M(xL-.4*m, yL)L(xL-.4*(m+.4), yL)}
DO(m,21){M(xL-.4*m,-yL)L(xL-.4*(m+.4),-yL)} fprintf(o,".1 W 0 0 0 RGB S\n");
//M(xL, yL)L(-8, yL)
//M(xL,-yL)L(-8,-yL) fprintf(o,"[.2 .2]0 setdash .12 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
//      system(    "open fig4b.eps");           //for macintosh
       system("epstopdf qexpMap.eps");
       system(    "open qexpMap.pdf");         //for  unix
//      system(    "xpdf fig4b.pdf &");         //for  unix
//      getchar(); system("killall Preview");   //for macintosh
}

Latex generator of labels

References

Categrory:Tetration