File:Tetma.jpg: Difference between revisions

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

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