File:Sqrt2figL45eT.png

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Complex map of the Abel function \(G\!=\!\mathrm{AuExp}_{\sqrt{2},5}\) of the exponential to base \(b\!=\!\sqrt{2}\) constructed at the fixed point \(L\!=\!4\) with Regular iteration and normalization \(G(5)\!=\!0\).

In the Mathematics of computation ^[1], this function is denoted as \(F_{4,5}^{-1}\).

The lines drawn correspond to \(u\!+\!\mathrm i v=G(x\!+\!\mathrm i y)\)

The map is used as Fig.9.7 at page 112 of book «Superfunctions», 2020 ^[2]^[3]
in order to confirm the successful construction and implementation of the growing Abelfunction \(\mathrm{AuExp}_{\sqrt{2},5}\) for the transfer function \(T\!=\!\exp_{\sqrt{2}}\) with the Regular iteration at the fixed point \(L\!=\!4\).

C++ generator of curves

Files ado.cin and conto.cin should be loaded to the working directory in order to compile the C++ 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 complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#include "conto.cin"

z_type f45E(z_type z){int n; z_type e,s; 
DB coeu[21]={1.,
0.44858743119526122890, .19037224679780675668, 
0.77829576536968278770e-1, 0.30935860305707997953e-1, 
0.12022125769065893274e-1, 0.45849888965617461424e-2,
0.17207423310577291102e-2, 0.63681090387985537364e-3,
0.23276960030302567773e-3, 0.84145511838119915857e-4,
0.30115646493706434120e-4, 0.10680745813035087964e-4,
0.37565713615564248453e-5, 0.13111367785052622794e-5,
0.45437916254218158081e-6, 0.15642984632975371803e-6,
0.53523276400816416929e-7, 0.18207786280204973113e-7,  
0.61604764947389226744e-8, 0.2e-8};
       e=exp(.32663425997828098238*(z-1.11520724513161));      
       s=coeu[20]; for(n=19;n>=0;n--) { s*=e; s+=coeu[n]; } 
//     s=coeu[19]; for(n=18;n>=0;n--) { s*=e; s+=coeu[n]; } 
       return 4.+s*e;}

z_type F45E(z_type z){ DB b=sqrt(2.);
                       if(Re(z)<-1.) return f45E(z);
                       return exp(F45E(z-1.)*log(b));
               }

z_type f45L(z_type z){ int n; z_type e,s;
DB Uco[21]={1,  
-.44858743119526122890,        .21208912005491969757, 
-.10218436750697167872,        0.49698683037371830337e-1, 
-0.2430759032611955221e-1,     0.11933088396510860210e-1,
-0.587369764200886206e-2,      0.289686728710575713e-2,
-0.1430908106079253664e-2,     0.7076637148565759223e-3,
-0.3503296158729878e-3,        0.17357560046634138e-3,
-0.86061011929162626e-4,       0.426959089012891e-4,
-0.2119302906819844809e-4,     0.1052442259960e-4,
-0.52285174354086e-5,          0.259844999161e-5,
-0.129178211214818578e-5,      0.4e-6  };
       z-=4.;
       s=Uco[19]; for(n=18; n>=0; n--){ s*=z; s+=Uco[n]; }
//     s=Uco[20]; for(n=19; n>=0; n--){ s*=z; s+=Uco[n]; }
//      return log(s*z)/.32663425997828098238 +1.1152091357215375; 
       return log(s*z)/.32663425997828098238 +1.11520724513161;
 }

z_type F45L(z_type z){ DB b=sqrt(2.);
                       if(abs(z-4.)>.4) return F45L(log(z)/log(b))+1. ;
                        return f45L(z);
               }
// #include"sqrt2f45E.cin"

main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
int M=501,M1=M+1;
int N=403,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("sqrt2figL45e.eps","w");  ado(o,202,202);
fprintf(o,"101 101 translate\n 10 10 scale\n");
DO(m,M1) X[m]=-10.+.04*(m-.5);
//DO(n,N1) Y[n]=-10.+.04*(n-.5);
 DO(n,200) Y[n]=sinh(3.*(n-200.5)/200.);
        Y[200]=-.0001;
        Y[201]= .0001;
for(n=202;n<N1;n++) Y[n]=sinh(3.*(n-200.5-2)/200.);
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,"1 setlinejoin 2 setlinecap\n");
fprintf(o," .006 W 0 0 0 RGB S\n");
// z_type tm,tp,F[M1*N1];
DO(m,M1)DO(n,N1){      g[m*N1+n]=9999;
                       f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; 
DO(n,N1){y=Y[n]; z=z_type(x,y);        
//              c=F45E(z);
               c=F45L(z);
               p=Re(c); 
               q=Im(c);
               if(p>-15. && p<15. && q>-15. && q<15.
//              && fabs(p)>1.e-14
//              && fabs(q)>1.e-14
               ) { g[m*N1+n]=p; f[m*N1+n]=q;}
       }}
p=2.5; q=.8;
for(m=-10;m<10;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<11;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 .9 0 0 RGB S\n");
for(m=0;m<10;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 .9 RGB S\n");
for(m= 1;m<11;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".03 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,".03 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0.  ),-2*p,2*p); fprintf(o,".03 W .5 0 .5 RGB S\n");
for(m=-11;m<16;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");
// #include "plofu.cin"
 M(-10,0)L(4,0)fprintf(o,"0 setlinecap .04 W 1 1 1 RGB S\n");
for(n=0;n<28;n++){ M(4-.5*(n+.2),0) L(4-.5*(n+.4),0) } fprintf(o,".06 W 1 .5 0 RGB S\n");
for(n=0;n<28;n++){ M(4-.5*(n+.7),0) L(4-.5*(n+.9),0) } fprintf(o,".06 W 0 .5 1 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
       system("epstopdf sqrt2figL45e.eps"); 
       system(    "open sqrt2figL45e.pdf"); //for macintosh
}

Latex generator of labels

%

 % Copyleft 2012 by Dmitrii Kouznetsov %<br>
 \documentclass[12pt]{article} %<br>
 \usepackage{geometry} %<br>
 \usepackage{graphicx} %<br>
 \usepackage{rotating} %<br>
 \paperwidth 1050pt %<br>
 \paperheight 1040pt %<br>
 \topmargin -98pt %<br>
 \oddsidemargin -99pt %<br>
 \textwidth 1100pt %<br>
 \textheight 1100pt %<br>
 \pagestyle {empty} %<br>
 \newcommand \sx {\scalebox} %<br>
 \newcommand \rot {\begin{rotate}} %<br>
 \newcommand \ero {\end{rotate}} %<br>
 \newcommand \ing {\includegraphics} %<br>
 \begin{document} %<br>
 \sx{5}{ \begin{picture}(208,205) %<br>
% \put(6,5){\ing{arctaniacontour}} %<br>
% \put(6,5){\ing{sqrt2figf45e}} %<br>
 \put(6,5){\ing{sqrt2figL45e}} %<br>
 \put(2,203.4){\sx{.7}{$y$}} %<br>
 \put(2,184){\sx{.6}{$8$}} %<br>
 \put(2,164){\sx{.6}{$6$}} %<br>
 \put(2,144){\sx{.6}{$4$}} %<br>
 \put(2,124){\sx{.6}{$2$}} %<br>
 \put(2,104){\sx{.6}{$0$}} %<br>
 \put(-3,84){\sx{.6}{$-2$}} %<br>
 \put(-3,64){\sx{.6}{$-4$}} %<br>
 \put(-3,44){\sx{.6}{$-6$}} %<br>
 \put(-3,24){\sx{.6}{$-8$}} %<br>
 \put(-2,00){\sx{.6}{$-10$}} %<br>
 \put( 22,0){\sx{.6}{$-8$}} %<br>
 \put( 42,0){\sx{.6}{$-6$}} %<br>
 \put( 62,0){\sx{.6}{$-4$}} %<br>
 \put( 82,0){\sx{.6}{$-2$}} %<br>
 \put(106,0){\sx{.6}{$0$}} %<br>
 \put(126,0){\sx{.6}{$2$}} %<br>
 \put(146,0){\sx{.6}{$4$}} %<br>
 \put(166,0){\sx{.6}{$6$}} %<br>
 \put(186,0){\sx{.6}{$8$}} %<br>
 \put(203,0){\sx{.7}{$x$}} %<br>
%<br>
 \put(020,103.5){\sx{.99}{\bf cut}} %<br>
 \put(182,103.5){\sx{.99}{$v\!=\!0$}} %<br>
%<br>  
\put(97,200){\rot{-73}\sx{.99}{$u\!=\!4$}\ero}%<br>
\put(175,182){\rot{12}\sx{.99}{$u\!=\!3.4$}\ero}%<br>
\put(182,153){\rot{-21}\sx{.99}{$u\!=\!3$}\ero}%<br>
\put(185,55){\rot{21}\sx{.99}{$u\!=\!3$}\ero}%<br>
\put(174,25){\rot{-13}\sx{.99}{$u\!=\!3.4$}\ero}%<br>
\put(103.4,15){\rot{71}\sx{.99}{$u\!=\!4$}\ero}%<br>
\put(074.6,29){\rot{42}\sx{.99}{$u\!=\!4.2$}\ero}%<br>
\put(059,60){\rot{7}\sx{.99}{$u\!=\!4.4$}\ero}%<br>
%<br>
\put(160,189){\rot{-60}\sx{.99}{$v\!=\!1$}\ero}%<br>
\put(147,6){\rot{42}\sx{.99}{$v\!=\!-1$}\ero}%<br>
%<br>
\put(42,163){\rot{64}\sx{.99}{$v\!=\!1$}\ero}%<br>
\put(113,122){\rot{65}\sx{.99}{$v\!=\!2$}\ero}%<br>
 \end{picture} %<br>
 } %<br>
 \end{document}

References

↑ http://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf D.Kouznetsov, H.Trappmnn. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, v.271, July 2010, p.1727-1756.
↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics - Lambert Academic Publishing, 2020/7/28
↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.

Keywords

«Abel function», «Abelexponential», «Abelfunction», «AuExp», «Base sqrt2», «BaseSqrt2», «Regular iteration», «SuExp», «Superfunction», «Superfunctions», «Table of superfunctions»,

«Суперфункции»,

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[1] ttp://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf D.Kouznetsov, H.Trappmnn. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, v.271, July 2010, p.1727-1756.

[bookA-2] ttps://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics - Lambert Academic Publishing, 2020/7/28

[bookM-3] ttps://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.

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