File:Expitemap.jpg

File
File history
File usage
Metadata

Size of this preview: 385 × 599 pixels. Other resolutions: 154 × 240 pixels | 1,825 × 2,841 pixels.

Original file ‎(1,825 × 2,841 pixels, file size: 1.88 MB, MIME type: image/jpeg)

Fig.15.5 at page 214 of book «Superfunctions»^[1], 2020:

Complex maps of iterates of natural exponent (exponential); \(\ u\!+\!\mathrm i v=\exp^n(x\!+\!\mathrm i y)\)

The maps appear also in the Russian version «Суперфункции» (2014) ^[2], Рис.15.5, стр. 213.

The two maps at the top refers to \(n\!=\!1\) and \(n\!=\!-1\). They correspond to \(\exp^1\!=\!\exp\) and \(\exp^{-1}\!=\!\ln\).

For \(n=0.5\), the detailed map of the Square root of exponential is loaded as https://mizugadro.mydns.jp/t/index.php/File:QexpMap1263x1243.png

Case \(n=0\) (identity function) is not presented as obvious.

Routines to evaluate the non-integer iterates of exponential are described also in the Vladikavkaz Mathematical Journal ^[3].

The iterates of exponential are expressed through the natural tetration \(\mathrm{tet}\) and arctetration \(\mathrm{tet}^{-1}\!=\!\mathrm{ate}\): \[ \exp^n(z)=\mathrm{tet}\big(n+\mathrm{ate}(z)\big) \] In this expression, number \(n\) of the iterate has no need to be integer.

The C++ implementation of functions tet and ate are loaded as fsexp.cin and fslog.cin.
The titles of these articles may look not so good. Names tet.cin and ate.cin (not loaded yet) are reserved for the optimized versions of these routines. The performance of the primary approximations can be significantly improved. This improvement can be achieved by the cost of reduction of region of the overlapping of the ranges of their validities. This overlapping is necessary for the numerical testing of the primary approximations of these functions.

C++ generator of the First map

/* As the codes to generate the maps are very similar, I load the only generator of the first map.
Files ado.cin, conto.cin, fsexp.cin, fslog.cin should be loaded 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 "fsexp.cin"
 #include "fslog.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
 int M=401,M1=M+1;
 int N=201,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("exp10map.eps","w");  ado(o,802,402);
 fprintf(o,"401 1 translate\n 100 100 scale\n");
 fprintf(o,"1 setlinejoin 2 setlinecap\n");
 DO(m,M1) X[m]=-4.+.02*(m-.5);
 DO(n,N1) Y[n]=0.+.02*(n-.5); 
// for(n=0;n<N1;n++) Y[n]=sinh((2./200.)*(n-200.5));
 for(m=-4;m<5;m++) {M(m,0)L(m,4)}
 for(n=0;n<5;n++) {M(  -4,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
 //fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n");
 DO(m,M1)DO(n,N1){      g[m*N1+n]=999;
                        f[m*N1+n]=999;}
 DO(m,M1){x=X[m]; printf("x=%6.3f\n",x);
 DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.019)
       { //     c=exp(z);
        c=FSEXP(1.+FSLOG(z));
         p=Re(c); q=Im(c);
        // if(p>-12 && p<12 && fabs(q)>1.e-12) 
        g[m*N1+n]=p;
        // if(q>-12 && q<12 && fabs(q)>1.e-12) 
        f[m*N1+n]=q;
       }
        }}
 fprintf(o,"1 setlinejoin 1 setlinecap\n");
 p=2.;q=1;
 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.  ),-p,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");

conto(o,f,w,v,X,Y,M,N, 1.3372357014306895, -p,p);fprintf(o,".005 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, .31813150520476413, -p,p);fprintf(o,".005 W 0 0 0 RGB S\n");

 fprintf(o,"0 setlinejoin 0 setlinecap\n");
 fprintf(o,"showpage\n");
 fprintf(o,"%c%cTrailer\n",'%','%');
 fclose(o);
 free(f);
 free(g);
 free(w);
       system("epstopdf exp10map.eps"); 
       system(    "open exp10map.pdf"); //for macintosh
       getchar(); system("killall Preview"); // For macintosh
 }

Latex combiner

\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{geometry}
\paperwidth 438px
%\paperheight 134px 
\paperheight 682px 
\topmargin -107pt
\oddsidemargin -84pt
\textheight 800px
\pagestyle{empty}
\begin{document}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}

\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}

\newcommand \LeLa {
\put(-24,384){\sx{4}{$y$}}
\put(-24,288){\sx{4}{$3$}}
\put(-24,188){\sx{4}{$2$}}
\put(-24,88){\sx{4}{$1$}}
\put(-24,-12){\sx{4}{$0$}}
}
\newcommand \DoLa {
\put(70,-36){\sx{3.8}{$-3$}}
\put(170,-36){\sx{3.8}{$-2$}}
\put(270,-36){\sx{3.8}{$-1$}}
\put(390,-36){\sx{3.8}{$0$}}
\put(490,-36){\sx{3.8}{$1$}}
\put(590,-36){\sx{3.8}{$2$}}
\put(690,-36){\sx{3.8}{$3$}}
\put(786,-36){\sx{3.8}{$x$}}
}
%\begin{figure}%[H]
~
%\sx{.294}{\begin{picture}(802,402)
\sx{.26}{\begin{picture}(802,402) 
\put(0,0){\ing{exp10map}} \LeLa \put(20,342){\sx{7}{$n\!=\!1$}} 
\put(46,300){\rot{0}\sx{5}{$v\!=\!0$} \ero}
\put(50,144){\rot{0}\sx{5}{$u\!=\!0$} \ero}
\put(46,-13){\rot{0}\sx{5}{$v\!=\!0$} \ero}
\put(260,220){\rot{90}\sx{4.5}{$u\!=\!-0.2$} \ero}
\put(424,224){\rot{90}\sx{5}{$u\!=\!-1$} \ero} 
\put(244,4){\rot{56}\sx{4.5}{$u\!=\!0.2$} \ero} 
\put(404,4){\rot{56}\sx{5}{$u\!=\!1$} \ero}
\put(474,4){\rot{56}\sx{5}{$u\!=\!2$} \ero} 
\end{picture}} ~ 
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm10map}} \put(20,340){\sx{7}{$n\!=\!-1$}} 

\put(30,144){\rot{56} \sx{5}{$u\!=\!1.4$}\ero} %%%%%%
\put(100,114){\rot{55} \sx{5}{$u\!=\!1.2$}\ero} %%%%%%
\put(160,102){\rot{54} \sx{5}{$u\!=\!1$}\ero} 

\put(2,42){\rot{-8} \sx{5}{$v\!=\!3$}\ero}
\put(546,196){\rot{56} \sx{5}{$v\!=\!1$}\ero}
\put(580,162){\rot{45} \sx{5}{$v\!=\!0.8$}\ero} 
\put(610,122){\rot{34} \sx{5}{$v\!=\!0.6$}\ero} 
\put(626,79){\rot{22} \sx{5}{$v\!=\!0.4$}\ero} 
\put(640,34){\rot{11} \sx{5}{$v\!=\!0.2$}\ero} 
\put(641,-12){\rot{0} \sx{5}{$v\!=\!0$}\ero} 
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp09map}} \LeLa  \put(20,340){\sx{7}{$n\!=\!0.9$}}
\put(50,202){\rot{20}\sx{5}{$v\!=\!0$} \ero}
\put(400,182){\rot{56}\sx{5}{$v\!=\!1$} \ero} 
\put(10,122){\sx{5}{\bf cut}} 
\put(178,2){\rot{54}\sx{5}{$u\!=\!0$} \ero}
\put(430,146){\rot{7}\sx{5}{$u\!=\!0$} \ero}
\put(432, 46){\rot{43}\sx{5}{$u\!=\!1$} \ero}
\put(482, 10){\rot{52}\sx{5}{$u\!=\!2$} \ero}
 \end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm09map}}  \put(20,340){\sx{7}{$n\!=\!-0.9$}} 
\put(180,198){\rot{27} \sx{5}{$u\!=\!1$} \ero}
\put(270,14){\rot{47} \sx{5}{$u\!=\!0$} \ero}
\put(10,122){\sx{5}{\bf cut}} 
\put(606,250){\rot{48} \sx{5}{$v\!=\!1$} \ero} 
\put(636,134){\rot{29} \sx{5}{$v\!=\!0.6$} \ero}
\put(644,84){\rot{19} \sx{5}{$v\!=\!0.4$} \ero}
\put(648,32){\rot{10} \sx{5}{$v\!=\!0.2$} \ero}
\put(649,-13){\rot{0} \sx{5}{$v\!=\!0$}\ero} 
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp08map}} \LeLa  \put(20,340){\sx{7}{$n\!=\!0.8$}}
\put(168,198){\rot{35}\sx{5}{$v\!=\!0$} \ero}
\put(390,182){\rot{66}\sx{5}{$v\!=\!1$} \ero} 
\put(430,146){\rot{13}\sx{5}{$u\!=\!0$} \ero}
\put(10,122){\sx{5}{\bf cut}} 
\put(40,0){\rot{46}\sx{3.8}{$u\!=\!-0.2$} \ero}
\put(242,2){\rot{54}\sx{5}{$u\!=\!0$} \ero}
\put(432, 42){\rot{46}\sx{5}{$u\!=\!1$} \ero}
\put(488, 1){\rot{59}\sx{5}{$u\!=\!2$} \ero}
 \end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm08map}}  \put(20,340){\sx{7}{$n\!=\!-0.8$}} 
\put(184,192){\rot{27} \sx{5}{$u\!=\!1$} \ero}
\put(10,122){\sx{5}{\bf cut}} 
\put(230,36){\rot{34} \sx{5}{$u\!=\!0$} \ero}
\put(6,1){\rot{52} \sx{4}{$u\!=\!0.4$} \ero}
\put(166,3){\rot{54} \sx{4}{$u\!=\!0.2$} \ero}
\put(506,234){\rot{53} \sx{4}{$v\!\approx\!1.337$} \ero} 
\put(606,220){\rot{42} \sx{5}{$v\!=\!1$} \ero} 
\put(636,120){\rot{26} \sx{5}{$v\!=\!0.6$} \ero}
\put(644,76){\rot{16} \sx{5}{$v\!=\!0.4$} \ero}
\put(648,30){\rot{8} \sx{5}{$v\!=\!0.2$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp05map}} \LeLa  \put(20,340){\sx{7}{$n\!=\!0.5$}} 
\put(360,294){\rot{11} \sx{5}{$u\!=\!-2$} \ero} 
\put(382,222){\rot{22} \sx{5}{$u\!=\!-1$} \ero} 
\put(420,146){\rot{38} \sx{5}{$u\!=\!0$} \ero} 
\put(490,99){\rot{54} \sx{5}{$u\!=\!1$} \ero} 
\put(550,82){\rot{58} \sx{5}{$u\!=\!2$} \ero}
\put(600,66){\rot{62} \sx{5}{$u\!=\!3$} \ero}
\put(10,122){\sx{5}{\bf cut}} 
\put(680,90){\rot{-16} \sx{5}{$v\!=\!3$} \ero}
\put(680,24){\rot{-8} \sx{5}{$v\!=\!1$} \ero}
\put(210,154){\rot{78} \sx{5}{$v\!=\!0$} \ero} 
\end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm05map}}  \put(20,340){\sx{7}{$n\!=\!-0.5$}}  
\put(602,386){\rot{-61} \sx{5}{$u\!=\!2$} \ero}
\put(382,302){\rot{-41} \sx{5}{$u\!=\!1$} \ero}
\put(20,272){\rot{-6} \sx{4.5}{$u\!\approx\!0.318$} \ero}
\put(82,206){\rot{-3} \sx{5}{$u\!=\!0$} \ero}
\put(10,122){\sx{5}{\bf cut}}
\put(486,298){\rot{42} \sx{5}{$v\!=\!2$} \ero}
\put(680,190){\rot{19} \sx{5}{$v\!=\!1$} \ero}
\put(680,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp02map}}  \put(20,340){\sx{7}{$n\!=\!0.2$}}  \LeLa %\DoLa
\put(353,224){\rot{60} \sx{5}{$u\!=\!-1$} \ero}
\put(452,224){\rot{66} \sx{5}{$u\!=\!0$} \ero}
\put(530,200){\rot{71} \sx{5}{$u\!=\!1$} \ero}
\put(602,174){\rot{73} \sx{5}{$u\!=\!2$} \ero}
\put(10,122){\sx{5}{\bf cut}}
\put(680,328){\rot{-15} \sx{5}{$v\!=\!5$} \ero}
\put(680,260){\rot{-11} \sx{5}{$v\!=\!4$} \ero}
\put(680,194){\rot{-10} \sx{5}{$v\!=\!3$} \ero}
\put(680,125){\rot{-7} \sx{5}{$v\!=\!2$} \ero}
\put(680,56){\rot{-3} \sx{5}{$v\!=\!1$} \ero}
\put(680,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\put(200,240){\rot{-31} \sx{4.6}{$v\!\approx\!1.337$} \ero}
\end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm02map}}  \put(20,340){\sx{7}{$n\!=\!-0.2$}}  %\DoLa 
\put(158,312){\rot{-57} \sx{5}{$u\!=\!-1$} \ero}
\put(302,326){\rot{-66} \sx{5}{$u\!=\!0$} \ero}
\put(436,346){\rot{-71} \sx{5}{$u\!=\!1$} \ero}
\put(566,388){\rot{-74} \sx{5}{$u\!=\!2$} \ero}
\put(10,122){\sx{5}{\bf cut}}
%\put(680,348){\rot{5} \sx{5}{$v\!=\!3$} \ero}
\put(680,268){\rot{9} \sx{5}{$v\!=\!2$} \ero}
\put(470,132){\rot{8} \sx{4.6}{$v\!\approx\!1.337$} \ero}
%\put(470,128){\rot{4.6} \sx{5}{$v\!\approx\!1.3372357$} \ero}
\put(680,124){\rot{4} \sx{5}{$v\!=\!1$} \ero}
\put(680,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp01map}}  \put(20,340){\sx{7}{$n\!=\!0.1$}}  \LeLa \DoLa
\put(340,228){\rot{74} \sx{5}{$u\!=\!-1$} \ero}
\put(438,228){\rot{79} \sx{5}{$u\!=\!0$} \ero}
\put(526,214){\rot{81} \sx{5}{$u\!=\!1$} \ero}
\put(614,200){\rot{83} \sx{5}{$u\!=\!2$} \ero}
\put(10,122){\sx{5}{\bf cut}}
\put(680,318){\rot{-6} \sx{5}{$v\!=\!4$} \ero}
\put(680,237){\rot{-5} \sx{5}{$v\!=\!3$} \ero}
\put(680,154){\rot{-4} \sx{5}{$v\!=\!2$} \ero}
\put(680,72){\rot{-3} \sx{5}{$v\!=\!1$} \ero}

\put(114,210){\rot{-19} \sx{4.6}{$v\!\approx\!1.337$} \ero}
\end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm01map}}  \put(20,340){\sx{7}{$n\!=\!-0.1$}}  \DoLa 
\put(228,330){\rot{-74} \sx{5}{$u\!=\!-1$} \ero}
\put(346,352){\rot{-80} \sx{5}{$u\!=\!0$} \ero}
\put(462,366){\rot{-82} \sx{5}{$u\!=\!1$} \ero}
\put(576,386){\rot{-81} \sx{5}{$u\!=\!2$} \ero}
\put(10,122){\sx{5}{\bf cut}}
\put(680,348){\rot{5} \sx{5}{$v\!=\!3$} \ero}
\put(680,224){\rot{4} \sx{5}{$v\!=\!2$} \ero}
\put(470,128){\rot{4} \sx{4.6}{$v\!\approx\!1.337$} \ero}
%\put(470,128){\rot{4.6} \sx{5}{$v\!\approx\!1.3372357$} \ero}
\put(680,104){\rot{3} \sx{5}{$v\!=\!1$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}

\end{document}

References

↑ https://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862
https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3
http://mizugadro.mydns.jp/BOOK/202.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
↑ https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.
↑ http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

Keywords

«Abel function», «Complex map», «Exponential», «fsexp.cin», «fslog.cin», «Iterate», «Superfunction», «Superfunctions», «Tetration»,

«Суперфунцкии»,

File history

Click on a date/time to view the file as it appeared at that time.

	Date/Time	Thumbnail	Dimensions	User	Comment
current	06:12, 1 December 2018		1,825 × 2,841 (1.88 MB)	Maintenance script (talk \| contribs)	Importing image file

You cannot overwrite this file.

File usage

There are no pages that use this file.

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.

If the file has been modified from its original state, some details may not fully reflect the modified file.

Horizontal resolution	300 dpi
Vertical resolution	300 dpi

[1] ttps://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862
https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3
http://mizugadro.mydns.jp/BOOK/202.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.

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

[3] ttp://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

[1]

[2]

[3]