File:Vladi03.jpg

Complex map of asymptotic approximation lima of natural tetration and its agreement fifi.

Top:

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

Bottom:

levels $D=1,2,4,6,8,10,12,14 ~ ~ $ are drawn. Level $D=1$ is drawn with thick line.

Usage: this is figure 14.9 of the book Суперфункции (2014, In Russian) ; the English version is in preparation in 2015.

First time published in the Vladikavkaz Matehmatical Journal .

C++ generator of the top map
typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)

//#include "superex.cin" int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; z_type Zo=z_type(.31813150520476413, 1.3372357014306895); z_type Zc=z_type(.31813150520476413,-1.3372357014306895);
 * 1) include "fsexp.cin"
 * 2) include "conto.cin"

int M=400,M1=M+1; int N=201,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("figfima.eps","w");ado(o,0,0,202,92); FILE *o;o=fopen("vladi03a.eps","w");ado(o,202,82); fprintf(o,"101 11 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-8.+.04*m; DO(n,N1) Y[n]= 0.+.025*n;

for(m=-8;m<9;m++){M(m,0)L(m,5)} for(n= 0;n<6;n++){M(-8,n)L(8,n)} fprintf(o,".006 W 0 0 0 RGB S\n");

DO(m,M1)DO(n,N1){      g[m*N1+n]=9999; f[m*N1+n]=9999; } //z_type F[M1*N1]; z_type F[81002]; DO(m,M1){x=X[m]; printf("50 run at x=%6.3f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); c=fima(z); p=Re(c); q=Im(c); if(p>-9 && p<9) g[m*N1+n]=p; if(q>-9 && q<9) f[m*N1+n]=q; }}

p=1;q=.5; conto(o,g,w,v,X,Y,M,N, ( Re(Zo) ),-q,q); fprintf(o,".1 W 1 .5 1 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( Im(Zo) ),-q,q); fprintf(o,".1 W .2 1 .4 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (-Im(Zo) ),-q,q); fprintf(o,".1 W .4 1 .2 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf vladi03a.eps"); system(   "open vladi03a.pdf");// macintosh //getchar; system("killall Preview");//macintosh }
 * 1) include"plofu.cin"

C++ generator of the virrin map
typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)

//#include "superex.cin" int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; //z_type Zo=z_type(.31813150520476413, 1.3372357014306895); //z_type Zc=z_type(.31813150520476413,-1.3372357014306895);
 * 1) include "fsexp.cin"
 * 2) include "conto.cin"

int M=400,M1=M+1; int N=125,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("figfimaE.eps","w");ado(o,0,0,202,70); FILE *o;o=fopen("vladi03b.eps","w");ado(o,202,72); fprintf(o,"101 11 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-8.+.04*m; DO(n,N1) Y[n]= 0.+.04*n;

for(m=-8;m<9;m++) {M( m,0)L(m,5)} for(n= 0;n<6;n++) {M(-8,n)L(8,n)} fprintf(o,".006 W 0 0 0 RGB S\n");

DO(m,M1)DO(n,N1){      g[m*N1+n]=9999; f[m*N1+n]=9999; } DO(m,M1){x=X[m]; printf("50 run at x=%6.3f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); c=fima(z)-exp(fima(z-1.)); p=Re(-log(c))/log(10.); if(p>-999 && p<999) g[m*N1+n]=p; }} fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf vladi03b.eps"); system(   "open vladi03b.pdf"); // for linux //getchar; system("killall Preview");// for macintosh }
 * 1) include"plodi.cin"

Latex combiner
\documentclass[12pt]{article} \usepackage{graphicx} \usepackage{rotating} \usepackage{geometry} \paperwidth 420px \paperheight 310px \topmargin -112pt \oddsidemargin -90pt \pagestyle{empty} \begin{document} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \fimax { \put(17,52){\sx{.44}{$y$}} %\put(17,55){\sx{.58}{$5$}} \put(17,43){\sx{.44}{$4$}} \put(17,33){\sx{.44}{$3$}} \put(17,23){\sx{.44}{$2$}} \put(17,13){\sx{.44}{$1$}} \put(17,03){\sx{.44}{$0$}} %\put(12, 5){\sx{.58}{$-1$}} \put(26,0){\sx{.44}{$-7$}} \put(36,0){\sx{.44}{$-6$}} \put(46,0){\sx{.44}{$-5$}} \put(56,0){\sx{.44}{$-4$}} \put(66,0){\sx{.44}{$-3$}} \put(76,0){\sx{.44}{$-2$}} \put(86,0){\sx{.44}{$-1$}} \put( 99,0){\sx{.44}{$0$}} \put(109,0){\sx{.44}{$1$}} \put(119,0){\sx{.44}{$2$}} \put(129,0){\sx{.44}{$3$}} \put(139,0){\sx{.44}{$4$}} \put(149,0){\sx{.44}{$5$}} \put(159,0){\sx{.44}{$6$}} \put(169,0){\sx{.44}{$7$}} \put(178,0){\sx{.44}{$x$}} } \hskip -40pt \sx{2.55}{\begin{picture}(200,60) %\put(0,6){\includegraphics{figfima}} \put(-1,-7){\includegraphics{vladi03a}} \multiput(27,38)(22.7,4){4}{\sx{.4}{\rot{-76} $v\!=\!\Im(L)$\ero}} \multiput(38,41)(22.7,4){4}{\sx{.4}{\rot{-76} $u\!=\!\Re(L)$\ero}} \multiput(77,15)(44.5,10.6){3}{\sx{.4}{\rot{0} $u\!=\!0$\ero}} \multiput(89,16.4)(44.5,10.6){2}{\sx{.4}{\rot{0} $v\!=\!1$\ero}} \fimax \end{picture}}

\hskip -40pt \sx{2.55}{\begin{picture}(200,62) %\put(0,6){\includegraphics{figfimaE}} \put(-1,-7){\includegraphics{vladi03b}} \fimax \put( 64,48){\sx{.6}{$D_{\rm fifi}>14$}} \put(151,8){\sx{.6}{$D_{\rm fifi}\!<\!1$}} \end{picture}} \end{document}