File:Vladi08.jpg

Agreement of approximations of tetration by elementary functions with the original representation through the Cauchi integral is shown in the left hand side map.

The similar map at the right hand side represents the comparison to the similar representation through the Cauchi integral, but the contour of integration is displaced for 1/2 to the left.

Levels $D=\mathrm{const}$ are drawn with step 2, but the exception is dome for level $D=1$, this level is drawn with thick lines.

$\displaystyle {\rm fse}(z)=\left\{ \begin{array}{ccrcccr} {\rm fima}(z)		&,&	 4.5&	\!<\! &\Im(z)			\\ {\rm tai}(z)		&,&	 1.5&	\!<\! &\Im(z)	&\!\le\!&	 4.5	\\ {\rm maclo}(z)		&,&	-1.5&	\le	&\Im(z) &\!\le\!&	 1.5	\\ {\rm tai}(z^{*})^{*}	&,&	-4.5&	\le	&\Im(z) &\!<\!&\!\! -1.5	\\ {\rm fima}(z^{*})^{*}	&,&	&		&\Im(z)	&\!<\!&\!\! -4.5 \end{array} \right.$

Mnemonics for the name of the approximation: Fast Super Exponent.

The agreement plotted is

$\displaystyle D=D_{8}(z)=-\lg\left( \frac {|\mathrm{fse}(z)-F_{4}(z)|} {|\mathrm{fse}(z)|+|F_{4}(z)|} \right) $

where $F_4$ denotes the 4th ackermann, id set, natural tetration evaluated through the Cauchi integral .

Usage: this is figure 14.11 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 first picture
Fsexp.cin, ado.cin, conto.cin, plodi.cin should be loaded in order to compile the code below

typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x 4.5) return fima(z); if(y> 1.5) return tai3(z); if(y>-1.5) return maclo(z); if(y>-4.5) return conj(tai3(conj(z))); return conj(fima(conj(z))); } //#include "superlo.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 "conto.cin"

int M=100,M1=M+1; int N=101,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("figcf4small.eps","w");ado(o,0,0,82,82); FILE *o;o=fopen("vladi08a.eps","w");ado(o,82,82); fprintf(o,"41 11 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-4.+.08*(m-.5); DO(n,N1) Y[n]=-1.+.08*(n-.5);

for(m=-3;m<4;m++) {    if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)}                 } for(n=0;n<7;n++) {     M(  -3,n)L(3,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("run at x=%6.3f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); c=fsexp(z); d=F4natu(z); p=abs(c-d);///(abs(c)+abs(d)); p=-log(p)/log(10.); //     p=Re(c); q=Im(c); if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8)      g[m*N1+n]=p; //     if(q>-999 && q<999 && fabs(q)> 1.e-8)                           f[m*N1+n]=q; }} //M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); //     system(    "gv fig08a.eps"); system("epstopdf vladi08a.eps"); system(   "open vladi08a.pdf"); // for linux //     getchar; system("killall Preview");// for macintosh }
 * 1) include"plodi.cin"

C++ generator of the second picture
vladif5c.cin also should be loaded

typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x 4.5) return fima(z); if(y> 1.5) return tai3(z); if(y>-1.5) return maclo(z); if(y>-4.5) return conj(tai3(conj(z))); return conj(fima(conj(z))); } //#include "superlo.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 "conto.cin"

int M=100,M1=M+1; int N=101,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("figcf5small.eps","w");ado(o,0,0,82,82); FILE *o;o=fopen("vladi08b.eps","w");ado(o,82,82); fprintf(o,"41 11 translate\n 10 10 scale\n");

DO(m,M1)X[m]=-4.+.08*(m-.5); DO(n,N1)Y[n]=-1.+.08*(n-.5);

for(m=-3;m<4;m++) {    if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)}                 } for(n= 0;n<7;n++) {    M(  -3,n)L(3,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("run at x=%6.3f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); c=fsexp(z); d=F5(z); p=abs(c-d);///(abs(c)+abs(d)); p=-log(p)/log(10.); //     p=Re(c); q=Im(c); if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p; //     if(q>-999 && q<999 && fabs(q)> 1.e-8)                   f[m*N1+n]=q; }}

//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf vladi08b.eps"); system(   "open vladi08b.pdf");//linux //getchar; system("killall Preview"); }
 * 1) include"plodi.cin"

Latex combiner
\documentclass[12pt]{article} \usepackage{graphicx} \usepackage{rotating} \usepackage{geometry} \paperwidth 378px %\paperheight 134px \paperheight 180px \topmargin -104pt \oddsidemargin -94pt \pagestyle{empty} \begin{document} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox}

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

\sx{2.2}{\begin{picture}(90,80) %\put(0,0){\includegraphics{figsexpF4}} %\put(0,0){\includegraphics{figcf4small}} \put(0,0){\includegraphics{vladi08a}} \put(5,68){\sx{.45}{$\Im(z)$}} \put(7,59){\sx{.5}{$5$}} \put(7,49){\sx{.5}{$4$}} \put(7,39){\sx{.5}{$3$}} \put(7,29){\sx{.5}{$2$}} \put(7,19){\sx{.5}{$1$}} \put(7, 9){\sx{.5}{$0$}} \put(70, 6){\sx{.45}{$\Re(z)$}} \put(60, 6){\sx{.5}{$2$}} \put(40, 6){\sx{.5}{$0$}} \put(17 ,6){\sx{.5}{$-\!2$}} \put(25,62){\sx{.5}{$D\!>\!14$}} \put(55,60){\sx{.5}{$12\!<\!D\!<\!14$}} \put(68,20){\sx{.5}{$D\!<\!1$}} \end{picture}} \sx{2.2}{\begin{picture}(80,80) %\put(0,0){\includegraphics{figsexpF5}} %\put(0,0){\includegraphics{figcf5small}} \put(0,0){\includegraphics{vladi08b}} \put(5,68){\sx{.45}{$\Im(z)$}} \put(7,59){\sx{.5}{$5$}} \put(7,49){\sx{.5}{$4$}} \put(7,39){\sx{.5}{$3$}} \put(7,29){\sx{.5}{$2$}} \put(7,19){\sx{.5}{$1$}} \put(7, 9){\sx{.5}{$0$}} \put(70, 6){\sx{.45}{$\Re(z)$}} \put(60, 6){\sx{.5}{$2$}} \put(40, 6){\sx{.5}{$0$}} \put(17 ,6){\sx{.5}{$-\!2$}} \put(31,56){\sx{.5}{$D\!>\!14$}} \put(68,20){\sx{.5}{$D\!<\!1$}} \end{picture}}

\end{document}