File:Ack4c.jpg

Summary

Title / Description	Complex map of tetration to base ${\displaystyle b=1.52598338517+0.0178411853321\,\mathrm {i} }$ is shown with lines of constant ${\displaystyle u}$ and lines of constant ${\displaystyle v}$, while ${\displaystyle u+\mathrm {i} v=\mathrm {tet} _{b}(x\!+\!\mathrm {i} y)}$
Citizendium author & Copyright holder	Copyright © Dmitrii Kouznetsov. See below for licence/re-use information.
Date created	2014 August
Country of first publication	Japan
Notes	I use this image in the article D.Kouznetsov. Holomorphic ackermanns. 2015, in preparation.
Other versions	http://mizugadro.mydns.jp/t/index.php/File:Ack4c.jpg
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C++ Generator of map

Files ado.cin, conto.cin, filog.cin, TetSheldonIma.inc, GLxw2048.inc should be loaded to 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++) #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 "filog.cin" z_type b=z_type( 1.5259833851700000, 0.0178411853321000); z_type a=log(b); z_type Zo=Filog(a); z_type Zc=conj(Filog(conj(a))); DB A=32.; z_type tetb(z_type z){ int k; DB t; z_type c, cu,cd; #include "GLxw2048.inc" int K=2048; //#include "ima6.inc" #include "TetSheldonIma.inc" z_type E[2048],G[2048]; DO(k,K){c=F[k]; E[k]=log(c)/a; G[k]=exp(a*c);} c=0.; z+=z_type(0.1196573712872846, 0.1299776198056910); DO(k,K){t=A*GLx[k];c+=GLw[k]*(G[k]/(z_type( 1.,t)-z)-E[k]/(z_type(-1.,t)-z));} cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) ); c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd; return c;} int main(){ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d; int M=601,M1=M+1; int N=461,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("tetsheldonmap.eps","w");ado(o,602,202); fprintf(o,"301 101 translate\n 10 10 scale\n"); DO(m,M1)X[m]=-30.+.1*(m); DO(n,200)Y[n]=-10.+.05*n; Y[200]=-.01; Y[201]= .01; for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-1.); for(m=-30;m<31;m++){if(m==0){M(m,-10.2)L(m,10.2)} else{M(m,-10)L(m,10)}} for(n=-10;n<11;n++){ M( -30,n)L(30,n)} fprintf(o,".008 W 0 0 0 RGB S\n"); DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;} DO(n,N1){y=Y[n]; for(m=295;m<305;m++) {x=X[m]; //printf("%5.2f\n",x); z=z_type(x,y); c=tetb(z); p=Re(c);q=Im(c); if(p>-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;} d=c; for(k=1;k<31;k++) { m1=m+k*10; if(m1>M) break; d=exp(a*d); p=Re(d);q=Im(d); if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;} } d=c; for(k=1;k<31;k++) { m1=m-k*10; if(m1<0) break; d=log(d)/a; p=Re(d);q=Im(d); if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;} } }} fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1;q=.5; for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".02 W 0 .6 0 RGB S\n"); for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".02 W .9 0 0 RGB S\n"); for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".02 W 0 0 .9 RGB S\n"); for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".08 W .9 0 0 RGB S\n"); for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".08 W .6 0 .6 RGB S\n"); for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf tetsheldonmap.eps"); system( "open tetsheldonmap.pdf"); getchar(); system("killall Preview"); }

Refrences

http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. (2009). Solution of F(z+1)=exp(F(z)) in the complex z-plane. Mathematics of Computation, 78: 1647-1670.

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. (In Russian), DOI:10.1090/S0025-5718-09-02188-7. Page 257, Figure 18.8.

http://mizugadro.mydns.jp/t/index.php/File:Ack4c.jpg

D.Kouznetsov. Holomorphic ackermanns. 2015, in preparation.