File:Expitemap.jpg
Complex maps of iterates of natural exponent;
$u\!+\!\mathrm i v=\exp^n(x\!+\!\mathrm i y)$
Usage: this is figure 15.5 of the book Суперфункции (2014, In Russian) [1]; the English version is in preparation in 2015.
Routines to evaluate the non–integer iterates are described also in the Vladikavkaz Matehmatical Journal [2].
Refereces
- ↑
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. - ↑ http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
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}
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 link to this file.
