File:QexpMap1263x1243.png

Complex map of function «square root of exponential», \( \varphi=\sqrt{\exp} \), improved version from article «Superfunctions and square root of factorial» ^[1]. In the article, the square root of exponential (shown in the map) is compared to the square root of factorial.

Function \( \varphi(z) = \mathrm{tet}(1/2+\mathrm{ate}(z)) = \exp^{1/2}(z)\)

appears as solution of equation

\( \varphi(\varphi(z))=\exp(z) \)

Existence of this function is shown in 1950 by Hellmuth Kneser ^[2].

Natural Tetration \(\mathrm{tet}\) and Arctetration \(\mathrm{ate}\!=\!\mathrm{tet}^{-1}\) are implemented in 2009-2010 ^[3][4] and described in book «Superfunctions» ^[5][6] (chapters 14 and 15), similar map appears in the left part of figure 15.5 at page 211 of the book for iterate \(n\!=\!0.5\ \).

C++ generator of curves

/* files ado.cin, conto.cin, fsexp.cin, fslog.cin shouls be loaded */

#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 "fsexp.cin"
#include "fslog.cin"

DB Lr=0.31813150520476413531;
DB Li=1.3372357014306894089;

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c; //d; //,cu,cd;
int M=601,M1=M+1;
int N=601,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("04.eps","w");  ado(o,1202,1202);
FILE *o;o=fopen("QexpMa.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-.5);
DO(n,N1) Y[n]=-6+.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=-12;m<13;m++){M(m,-6) L(m,6)  }
for(n=-12;n<13;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=F(z);
  c=FSEXP(.5+FSLOG(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>-99 && p<99 ){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");
 p=.4;q=.2;

for(m=-5;m<5;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".01 W 0 .6 0 RGB S\n");
for(m=0;m<5;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".01 W .8 0 0 RGB S\n");
for(m=0;m<5;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".01 W 0 0 .8 RGB S\n");
for(m= 1;m<11;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<11;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=-10;m<11;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,"0 setlinejoin 0 setlinecap\n");
M(-6.1, Li)L(Lr, Li) 
M(-6.1,-Li)L(Lr,-Li) 
fprintf(o,".034 W 1 1 1 RGB S\n");

for(m=0;m<80;m+=2) 
{DB x; x=Lr-.1*m;
M(x,Li); L(x-.1,Li);
if(x<-6.) break;
}
for(m=0;m<80;m+=2) 
{DB x; x=Lr-.1*m;
M(x,-Li); L(x-.1,-Li);
if(x<-6.) break;
}

fprintf(o,".054 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n");
fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o);  free(f); free(g); free(w);
      system("epstopdf QexpMa.eps"); 
      system(    "open QexpMa.pdf"); //for macintosh
getchar(); system("killall Preview"); //for macintosh
}

Latex generator of labels

\documentclass[12pt]{article} 
\usepackage{geometry} 
\usepackage{graphicx} 
\usepackage{rotating} 
\paperwidth 1268pt 
\paperheight 1248pt 
\topmargin -126pt 
\oddsidemargin -70pt 
\textwidth 1400pt 
\textheight 1400pt 
\pagestyle {empty} 
\newcommand \sx {\scalebox} 
\newcommand \rot {\begin{rotate}} 
\newcommand \ero {\end{rotate}} 
\newcommand \ing {\includegraphics} 
\parindent 0pt% <br> 
\pagestyle{empty} 
\begin{document} 
\begin{picture}(1260,1250) 
\put(50,24){\ing{QexpMa}} 
\put(24,1211){\sx{3.8}{$y$}}
\put(24,1113){\sx{3.8}{$5$}}
\put(24,1013){\sx{3.8}{$4$}}
\put(24,913){\sx{3.8}{$3$}}
\put(24,813){\sx{3.8}{$2$}}
\put(24,713){\sx{3.8}{$1$}}
\put(24,613){\sx{3.8}{$0$}}
\put(-8,514){\sx{3.8}{$-1$}}
\put(-8,414){\sx{3.8}{$-2$}}
\put(-8,314){\sx{3.8}{$-3$}}
\put(-8,214){\sx{3.8}{$-4$}}
\put(-8,114){\sx{3.8}{$-5$}}
\put(-8,15){\sx{3.8}{$-6$}}
\put(9,-9){\sx{3.8}{$-6$}}
\put(109,-9){\sx{3.8}{$-5$}}
\put(209,-9){\sx{3.8}{$-4$}}
\put(309,-9){\sx{3.8}{$-3$}}
\put(409,-9){\sx{3.8}{$-2$}}
\put(509,-9){\sx{3.8}{$-1$}}
\put(644,-9){\sx{3.8}{$0$}}
\put(744,-9){\sx{3.8}{$1$}}
\put(844,-9){\sx{3.8}{$2$}}
\put(944,-9){\sx{3.8}{$3$}}
\put(1044,-9){\sx{3.8}{$4$}}
\put(1144,-9){\sx{3.8}{$5$}}
\put(1235,-8){\sx{3.8}{$x$}}
\put(1158,836){\sx{4.6}{\rot{47}$u\!=\!10$\ero}}
\put(186,810){\sx{5}{\rot{30}$v\!=\!-1$\ero}}
\put(470,834){\sx{5}{\rot{74}$v\!=\!0$\ero}}
\put(254,750){\sx{5}{\bf cut}}
\put(1074,634){\sx{4.3}{\rot{-3}$v\!=\!1$\ero}}
\put(177,614){\sx{5}{$v\!=\!0$}}
\put(1070,590){\sx{4.3}{\rot{2}$v\!=\!-1$\ero}}
\put(392,528){\sx{4.5}{\rot{90}$u\!=\!-0.6$\ero}}
\put(602,564){\sx{5}{\rot{90}$u\!=\!0$\ero}}
\put(722,564){\sx{5}{\rot{90}$u\!=\!1$\ero}}
\put(254,480){\sx{5}{\bf cut}}
\put(186,412){\sx{5}{\rot{-30}$v\!=\!1$\ero}}
\put(136,255){\sx{5}{\rot{-16}$v\!=\!2$\ero}}
\end{picture}
\end{document}

References

Keywords

«Abelfunction», «ArcTetration», «Book», «BookMap», «Hellmuth Kneser», «Iterate», «Iteration half of exponential», «Iteration half of factorial», «Natural tetration», «Square root of exponential», «Square root of factorial», «Superfunction», «Superfunctions», «Tetration»,