File:Tetma.jpg

Summary

Licensing

   

This media, Tetma.jpg, is licenced under the Creative Commons CC0 1.0 Universal License

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Description

Tetration tet is shown with

lines ${\displaystyle u=\Re (\mathrm {tet} (x\!+\!\mathrm {i} y))=\mathrm {const} }$ and lines

${\displaystyle u=\Im (\mathrm {tet} (x\!+\!\mathrm {i} y))=\mathrm {const} }$

in the x,y plane. In order to simplify the incorporation to various texts, the labels of the lines and those of the axes are suppressed.

The tetration and its maps are described in the Mathematica of Computation: http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html 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.

The fast algorithm of evaluation of tetration is described in the Vladikavkaz mathematical Journal: http://www.ils.uec.ac.jp/~dima/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

The image is used at the front cover of the book about Superfunctions, the Book is in Russian and is available at
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
http://mizugadro.mydns.jp/BOOK/201.pdf
http://www.ils.uec.ac.jp/~dima/BOOK/201.pdf
http://mizugadro.mydns.jp/t/index.php/Суперфункции

C++ generator of the image

Files ado.cin, conto.cin, fslog.cin should be loaded into 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 "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=801,M1=M+1; int N=401,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("b271t0.eps","w"); ado(o,87,87); FILE *o;o=fopen("tetma.eps","w"); ado(o,1604,804); fprintf(o,"802 402 translate\n 100 100 scale\n"); DO(m,M1) X[m]=-8.+.02*(m-.5); DO(n,N1) Y[n]=-4.+.02*(n-.5); for(m=-8;m<17;m++) {M(m,-4)L(m,4)} for(n=-4;n<5;n++) {M( -8,n)L(8,n)} fprintf(o,"2 setlinecap .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]=9999; f[m*N1+n]=9999;} //DB b=sqrt(2); 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.)>.04) { c=FSEXP(z); // c=FSLOG(z); p=Re(c); q=Im(c); if(p>-9999 && p<9999 && fabs(q)>1.e-12) g[m*N1+n]=p; if(q>-9999 && q<9999 && fabs(q)>1.e-12) f[m*N1+n]=q; } }} fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=2.; q=1.1;; //#include "plofu.cin" p=2;q=1; for(m=-19;m<19;m++)for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".02 W 0 1 0 RGB S\n"); for(m=0;m<29;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q); fprintf(o,".01 W 1 0 0 RGB S\n"); for(m=0;m<29;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".01 W 0 0 1 RGB S\n"); for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".03 W 1 0 0 RGB S\n"); for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".03 W 0 0 1 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".03 W .8 0 .8 RGB S\n"); for(m=-31;m<32;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n"); fprintf(o,"0 setlinejoin 0 setlinecap\n"); fprintf(o,"showpage\n"); fprintf(o,"%cTrailer\n",'%'); fclose(o); system("epstopdf tetma.eps"); system( "open tetma.pdf"); //for macintosh getchar(); system("killall Preview"); // For macintosh