Difference between revisions of "File:Ack4c.jpg"

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  +
[[Complex map]] of [[tetration to Sheldon base]] $b\!=\!
−
Importing image file
  
  +
1.52598338517+0.0178411853321 i$.
  +
  +
$u\!+\!\mathrm i v=\mathrm{tet}_b(x\!+\!\mathrm i y)$
  +
  +
Another version of this image is loaded as http://mizugadro.mydns.jp/t/index.php/File:Tetsheldonmap03.png
  +
==[[C++]] Generator of map==
  +
Files
  +
[[ado.cin]],
  +
[[conto.cin]],
  +
[[filog.cin]],
  +
[[TetSheldonIma.inc]],
  +
[[GLxw2048.inc]]
  +
should be loaded to the working directory in order to compile the code below.
  +
<poem><nomathjax><nowiki>
  +
#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 "filog.cin"
  +
  +
z_type b=z_type( 1.5259833851700000, 0.0178411853321000);
  +
z_type a=log(b);
  +
z_type Zo=Filog(a);
  +
z_type Zc=conj(Filog(conj(a)));
  +
DB A=32.;
  +
  +
z_type tetb(z_type z){ int k; DB t; z_type c, cu,cd;
  +
#include "GLxw2048.inc"
  +
int K=2048;
  +
//#include "ima6.inc"
  +
#include "TetSheldonIma.inc"
  +
z_type E[2048],G[2048];
  +
DO(k,K){c=F[k]; E[k]=log(c)/a; G[k]=exp(a*c);}
  +
c=0.;
  +
z+=z_type(0.1196573712872846, 0.1299776198056910);
  +
DO(k,K){t=A*GLx[k];c+=GLw[k]*(G[k]/(z_type( 1.,t)-z)-E[k]/(z_type(-1.,t)-z));}
  +
cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) );
  +
cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) );
  +
c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd;
  +
return c;}
  +
  +
int main(){ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d;
  +
//int M=161,M1=M+1;
  +
int M=601,M1=M+1;
  +
int N=461,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("tetsheldonmap.eps","w");ado(o,602,202);
  +
fprintf(o,"301 101 translate\n 10 10 scale\n");
  +
DO(m,M1)X[m]=-30.+.1*(m);
  +
DO(n,200)Y[n]=-10.+.05*n;
  +
Y[200]=-.01;
  +
Y[201]= .01;
  +
for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-1.);
  +
for(m=-30;m<31;m++){if(m==0){M(m,-10.2)L(m,10.2)} else{M(m,-10)L(m,10)}}
  +
for(n=-10;n<11;n++){ M( -30,n)L(30,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(n,N1){y=Y[n];
  +
for(m=295;m<305;m++)
  +
{x=X[m]; //printf("%5.2f\n",x);
  +
z=z_type(x,y);
  +
c=tetb(z);
  +
p=Re(c);q=Im(c);
  +
if(p>-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
  +
d=c;
  +
for(k=1;k<31;k++)
  +
{ m1=m+k*10; if(m1>M) break;
  +
d=exp(a*d);
  +
p=Re(d);q=Im(d);
  +
if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
  +
}
  +
d=c;
  +
for(k=1;k<31;k++)
  +
{ m1=m-k*10; if(m1<0) break;
  +
d=log(d)/a;
  +
p=Re(d);q=Im(d);
  +
if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
  +
}
  +
}}
  +
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1;q=.5;
  +
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,".02 W 0 .6 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,".02 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,".02 W 0 0 .9 RGB S\n");
  +
for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".08 W .9 0 0 RGB S\n");
  +
for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 .9 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".08 W .6 0 .6 RGB S\n");
  +
for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 0 RGB S\n");
  +
// y= 0; for(m=0;m<260;m+=6) {x=-2.-.1*m; M(x,y) L(x-.1,y)}
  +
// fprintf(o,".07 W 1 .5 0 RGB S\n");
  +
// y= 0; for(m=3;m<260;m+=6) {x=-2-.1*m; M(x,y) L(x-.1,y)}
  +
// fprintf(o,".07 W 0 .5 1 RGB S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf tetsheldonmap.eps");
  +
system( "open tetsheldonmap.pdf");
  +
getchar(); system("killall Preview");
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] Generator of labels]==
  +
<poem><nomathjax><nowiki>
  +
\documentclass{amsproc}
  +
\usepackage{graphicx} % Use pdf, png, jpg, or eps§ with pdflatex; use eps in DVI mode\usepackage{amssymb}
  +
\usepackage{rotating}
  +
\usepackage{hyperref}
  +
\newcommand \be {\begin{eqnarray}}
  +
\newcommand \ee {\end{eqnarray} }
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rme {{\rm e}} %%makes the base of natural logarithms Roman font
  +
%\newcommand \rme {{e}} %%makes the base of natural logarithms Italics font; choose one of these
  +
\newcommand \rmi {{\rm i}} %%imaginary unity is always roman font
  +
\newcommand \ds {\displaystyle}
  +
\newcommand \bN {\mathbb{N}}
  +
\newcommand \bC {\mathbb{C}}
  +
\newcommand \bR {\mathbb{R}}
  +
\newcommand \cO {\mathcal{O}}
  +
\newcommand \cF {\mathcal{F}}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \nS {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
  +
\newcommand \pS {{~}~{~}}
  +
\newcommand \fac {\mathrm{Factorial}}
  +
\newcommand {\rf}[1] {(\ref{#1})}
  +
\newcommand{\iL}[1] {~\label{#1}\pS \rm[#1]\nS} %make the labels visible
  +
%\newcommand{\iL}[1] {\label{#1}} %make the labels invisible; choose one of these options
  +
\newcommand \eL[1] {\iL{#1}\ee}
  +
\newcommand \ing \includegraphics
  +
\newcommand \tet {\mathrm{tet}}
  +
  +
\usepackage{geometry}
  +
%\topmargin -94pt
  +
\topmargin -97pt
  +
\oddsidemargin -87pt
  +
\paperwidth 618pt
  +
%\paperheight 216pt
  +
\paperheight 214pt
  +
  +
\begin{document} %\title{AMS Proceedings Series Sample}
  +
  +
\newcommand \mapax {
  +
\put(2,206){\sx{1.2}{$y$}}
  +
\put(2,188){\sx{1.2}{$8$}}
  +
\put(2,168){\sx{1.2}{$6$}}
  +
\put(2,148){\sx{1.2}{$4$}}
  +
\put(2,128){\sx{1.2}{$2$}}
  +
\put(2,108){\sx{1.2}{$0$}}
  +
\put(-6,88){\sx{1.2}{$-2$}}
  +
\put(-6,68){\sx{1.2}{$-4$}}
  +
\put(-6,48){\sx{1.2}{$-6$}}
  +
\put(-6,28){\sx{1.2}{$-8$}}
  +
\put(-1,1){\sx{1.2}{$-30$}}
  +
\put( 49,1){\sx{1.2}{$-25$}}
  +
\put( 99,1){\sx{1.2}{$-20$}}
  +
\put(149,1){\sx{1.2}{$-15$}}
  +
\put(199,1){\sx{1.2}{$-10$}}
  +
\put(252,1){\sx{1.2}{$-5$}}
  +
\put(309,1){\sx{1.2}{$0$}}
  +
\put(329,1){\sx{1.2}{$2$}}
  +
\put(349,1){\sx{1.2}{$4$}}
  +
\put(369,1){\sx{1.2}{$6$}}
  +
\put(389,1){\sx{1.2}{$8$}}
  +
\put(407,1){\sx{1.2}{$10$}}
  +
\put(457,1){\sx{1.2}{$15$}}
  +
\put(507,1){\sx{1.2}{$20$}}
  +
\put(557,1){\sx{1.2}{$25$}}
  +
\put(607,1){\sx{1.2}{$x$}}
  +
}
  +
%\flushright{$b=1.5 ~$}
  +
%\sx{.586}
  +
{\begin{picture}(620,216) \mapax \put(10,10){\ing{tet15ma}} %%%%%%%%
  +
\put(25,108.4){\sx{1.4}{\bf cut}} \put(502,108.4){\sx{1.4}{$v\!=\!0$}}
  +
%
  +
\put(24,194){\sx{1.5}{$u\!+\!\mathrm i v \approx 2.306009391950\!+\! 1.081988656014\,\mathrm i$}}
  +
\put(24,19){\sx{1.5}{$u\!+\!\mathrm i v \approx 2.306009391950\!-\! 1.081988656014\,\mathrm i$}}
  +
%\put(20,16){\sx{1.5}{$u\!+\!\mathrm i v \approx 2.3\!-\!1.1\,\mathrm i$}}
  +
%
  +
\multiput(50,160)(143,10.4){4}{\sx{1.4}{$v\!=\!1$}}%%
  +
\multiput(48,129)(143,10.4){4}{\sx{1.4}{$v\!=\!0.8$}}%%
  +
\put(342,108){\sx{1.4}{$v\!=\!0$}}%%
  +
\multiput(46,56)(143,-10.4){4}{\sx{1.4}{$v\!=\!-1$}}%%
  +
  +
\put(341,96){\sx{1.4}{\rot{90}$u\!=\!2$\ero}}
  +
\put(381,96){\sx{1.4}{\rot{90}$u\!=\!3$\ero}}
  +
\end{picture}}%%%%%%%%%
  +
\end{document}
  +
\caption{$u\!+\!\mathrm i v\!=\!\mathrm{tet}_b(x\!+\!\mathrm i y)$ for
  +
$b\!=\!\sqrt{2}
  +
%\!\approx\! 1.41
  +
$,
  +
$b\!=\!\exp(1/\mathrm e)
  +
%\!\approx\! 1.44
  +
$, and
  +
$b\!=\!1.5~$
  +
\label{maps1}}
  +
\end{figure}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
==Refrences==
  +
  +
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html <br>
  +
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf<br>
  +
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf
  +
D.Kouznetsov. (2009). Solutions of F(z+1)=exp(F(z)) in the complex plane. Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7.
  +
  +
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. (In Russian)
  +
  +
D.Kouznetsov. Holomorphic ackermanns. 2015, in preparation.
  +
  +
[[Category:Book]]
  +
[[Category:BookPlot]]
  +
[[Category:Tetration]]
  +
[[Category:Sheldon base]]
  +
[[Category:Complex map]]
  +
[[Category:AMS]]
  +
[[Category:C++]]
  +
[[Category:Latex]]

Latest revision as of 08:28, 1 December 2018

Complex map of tetration to Sheldon base $b\!=\! 1.52598338517+0.0178411853321 i$.

$u\!+\!\mathrm i v=\mathrm{tet}_b(x\!+\!\mathrm i y)$

Another version of this image is loaded as http://mizugadro.mydns.jp/t/index.php/File:Tetsheldonmap03.png

C++ Generator of map

Files ado.cin, conto.cin, filog.cin, TetSheldonIma.inc, GLxw2048.inc 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 "filog.cin"

z_type b=z_type( 1.5259833851700000, 0.0178411853321000);
z_type a=log(b);
z_type Zo=Filog(a);
z_type Zc=conj(Filog(conj(a)));
DB A=32.;

z_type tetb(z_type z){ int k; DB t; z_type c, cu,cd;
#include "GLxw2048.inc"
int K=2048;
//#include "ima6.inc"
#include "TetSheldonIma.inc"
z_type E[2048],G[2048];
DO(k,K){c=F[k]; E[k]=log(c)/a; G[k]=exp(a*c);}
c=0.;
z+=z_type(0.1196573712872846, 0.1299776198056910);
DO(k,K){t=A*GLx[k];c+=GLw[k]*(G[k]/(z_type( 1.,t)-z)-E[k]/(z_type(-1.,t)-z));}
 cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) );
 cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) );
 c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd;
 return c;}

int main(){ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d;
 //int M=161,M1=M+1;
 int M=601,M1=M+1;
 int N=461,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("tetsheldonmap.eps","w");ado(o,602,202);
 fprintf(o,"301 101 translate\n 10 10 scale\n");
 DO(m,M1)X[m]=-30.+.1*(m);
 DO(n,200)Y[n]=-10.+.05*n;
         Y[200]=-.01;
         Y[201]= .01;
 for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-1.);
 for(m=-30;m<31;m++){if(m==0){M(m,-10.2)L(m,10.2)} else{M(m,-10)L(m,10)}}
 for(n=-10;n<11;n++){ M( -30,n)L(30,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(n,N1){y=Y[n];
          for(m=295;m<305;m++)
          {x=X[m]; //printf("%5.2f\n",x);
           z=z_type(x,y);
           c=tetb(z);
           p=Re(c);q=Im(c);
           if(p>-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
           d=c;
           for(k=1;k<31;k++)
                { m1=m+k*10; if(m1>M) break;
                d=exp(a*d);
                p=Re(d);q=Im(d);
                if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
                }
           d=c;
           for(k=1;k<31;k++)
                { m1=m-k*10; if(m1<0) break;
                d=log(d)/a;
                p=Re(d);q=Im(d);
                if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
                }
        }}

 fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1;q=.5;
 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,".02 W 0 .6 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,".02 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,".02 W 0 0 .9 RGB S\n");
 for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".08 W .9 0 0 RGB S\n");
 for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 .9 RGB S\n");
                    conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".08 W .6 0 .6 RGB S\n");
 for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 0 RGB S\n");
// y= 0; for(m=0;m<260;m+=6) {x=-2.-.1*m; M(x,y) L(x-.1,y)}
// fprintf(o,".07 W 1 .5 0 RGB S\n");
// y= 0; for(m=3;m<260;m+=6) {x=-2-.1*m; M(x,y) L(x-.1,y)}
// fprintf(o,".07 W 0 .5 1 RGB S\n");
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf tetsheldonmap.eps");
        system( "open tetsheldonmap.pdf");
        getchar(); system("killall Preview");
 }

Latex Generator of labels]


\documentclass{amsproc}
\usepackage{graphicx} % Use pdf, png, jpg, or eps§ with pdflatex; use eps in DVI mode\usepackage{amssymb}
\usepackage{rotating}
\usepackage{hyperref}
\newcommand \be {\begin{eqnarray}}
\newcommand \ee {\end{eqnarray} }
\newcommand \sx {\scalebox}
\newcommand \rme {{\rm e}} %%makes the base of natural logarithms Roman font
%\newcommand \rme {{e}} %%makes the base of natural logarithms Italics font; choose one of these
\newcommand \rmi {{\rm i}} %%imaginary unity is always roman font
\newcommand \ds {\displaystyle}
\newcommand \bN {\mathbb{N}}
\newcommand \bC {\mathbb{C}}
\newcommand \bR {\mathbb{R}}
\newcommand \cO {\mathcal{O}}
\newcommand \cF {\mathcal{F}}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \nS {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
\newcommand \pS {{~}~{~}}
\newcommand \fac {\mathrm{Factorial}}
\newcommand {\rf}[1] {(\ref{#1})}
\newcommand{\iL}[1] {~\label{#1}\pS \rm[#1]\nS} %make the labels visible
%\newcommand{\iL}[1] {\label{#1}} %make the labels invisible; choose one of these options
\newcommand \eL[1] {\iL{#1}\ee}
\newcommand \ing \includegraphics
\newcommand \tet {\mathrm{tet}}

\usepackage{geometry}
%\topmargin -94pt
\topmargin -97pt
\oddsidemargin -87pt
\paperwidth 618pt
%\paperheight 216pt
\paperheight 214pt

\begin{document} %\title{AMS Proceedings Series Sample}

\newcommand \mapax {
\put(2,206){\sx{1.2}{$y$}}
\put(2,188){\sx{1.2}{$8$}}
\put(2,168){\sx{1.2}{$6$}}
\put(2,148){\sx{1.2}{$4$}}
\put(2,128){\sx{1.2}{$2$}}
\put(2,108){\sx{1.2}{$0$}}
\put(-6,88){\sx{1.2}{$-2$}}
\put(-6,68){\sx{1.2}{$-4$}}
\put(-6,48){\sx{1.2}{$-6$}}
\put(-6,28){\sx{1.2}{$-8$}}
\put(-1,1){\sx{1.2}{$-30$}}
\put( 49,1){\sx{1.2}{$-25$}}
\put( 99,1){\sx{1.2}{$-20$}}
\put(149,1){\sx{1.2}{$-15$}}
\put(199,1){\sx{1.2}{$-10$}}
\put(252,1){\sx{1.2}{$-5$}}
\put(309,1){\sx{1.2}{$0$}}
\put(329,1){\sx{1.2}{$2$}}
\put(349,1){\sx{1.2}{$4$}}
\put(369,1){\sx{1.2}{$6$}}
\put(389,1){\sx{1.2}{$8$}}
\put(407,1){\sx{1.2}{$10$}}
\put(457,1){\sx{1.2}{$15$}}
\put(507,1){\sx{1.2}{$20$}}
\put(557,1){\sx{1.2}{$25$}}
\put(607,1){\sx{1.2}{$x$}}
}
%\flushright{$b=1.5 ~$}
%\sx{.586}
{\begin{picture}(620,216) \mapax \put(10,10){\ing{tet15ma}} %%%%%%%%
\put(25,108.4){\sx{1.4}{\bf cut}} \put(502,108.4){\sx{1.4}{$v\!=\!0$}}
%
\put(24,194){\sx{1.5}{$u\!+\!\mathrm i v \approx 2.306009391950\!+\! 1.081988656014\,\mathrm i$}}
\put(24,19){\sx{1.5}{$u\!+\!\mathrm i v \approx 2.306009391950\!-\! 1.081988656014\,\mathrm i$}}
%\put(20,16){\sx{1.5}{$u\!+\!\mathrm i v \approx 2.3\!-\!1.1\,\mathrm i$}}
%
\multiput(50,160)(143,10.4){4}{\sx{1.4}{$v\!=\!1$}}%%
\multiput(48,129)(143,10.4){4}{\sx{1.4}{$v\!=\!0.8$}}%%
\put(342,108){\sx{1.4}{$v\!=\!0$}}%%
\multiput(46,56)(143,-10.4){4}{\sx{1.4}{$v\!=\!-1$}}%%

\put(341,96){\sx{1.4}{\rot{90}$u\!=\!2$\ero}}
\put(381,96){\sx{1.4}{\rot{90}$u\!=\!3$\ero}}
\end{picture}}%%%%%%%%%
\end{document}
\caption{$u\!+\!\mathrm i v\!=\!\mathrm{tet}_b(x\!+\!\mathrm i y)$ for
$b\!=\!\sqrt{2}
%\!\approx\! 1.41
$,
$b\!=\!\exp(1/\mathrm e)
%\!\approx\! 1.44
$, and
$b\!=\!1.5~$
\label{maps1}}
\end{figure}
\end{document}

Refrences

http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. (2009). Solutions of F(z+1)=exp(F(z)) in the complex plane. Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7.

https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014. (In Russian)

D.Kouznetsov. Holomorphic ackermanns. 2015, in preparation.

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