File:Ack4c.jpg: Difference between revisions
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\put(312,720){\sx{1.8}{\rot{80}$y\!=\!\mathcal{A}(3,x)\!=\! A_{2,3}(x\!+\!3)\!-\!3$\ero}} | \put(312,720){\sx{1.8}{\rot{80}$y\!=\!\mathcal{A}(3,x)\!=\! A_{2,3}(x\!+\!3)\!-\!3$\ero}} | ||
\put(348,712){\sx{1.8}{\rot{63}$y\!=\!\mathcal{A}(2,x)\!=\! A_{2,2}(x\!+\!3)\!-\!3$\ero}} | \put(348,712){\sx{1.8}{\rot{63}$y\!=\!\mathcal{A}(2,x)\!=\! A_{2,2}(x\!+\!3)\!-\!3$\ero}} | ||
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Revision as of 14:11, 3 September 2014
Summary
| Title / Description
|
Complex map of tetration to base
is shown with lines of constant and lines of constant , while |
|---|---|
| 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 |
| Using this image on CZ
|
, then copy the code below to add this image to a Citizendium article, changing the size, alignment, and caption as necessary.
|
Please send email to manager A T citizendium.org .
Licensing
This media, Ack4c.jpg, is licenced under the Creative Commons Attribution 3.0 Unported License
You are free:
To Share — To copy, distribute and transmit the work; To Remix — To adapt the work.
Under the following conditions:
Attribution — You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).
For any reuse or distribution, you must make clear to others the licence terms of this work (the best way to do this is with a link to this licence's web page). Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this licence impairs or restricts the author's moral rights.
Read the full licence.
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"); }
Latex generator of labels
\documentclass[12pt]{article} \paperwidth 640px \paperheight 1006px \textwidth 1394px \textheight 1300px \topmargin -104px \oddsidemargin -92px \usepackage{graphics} \usepackage{rotating} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \begin{document} {\begin{picture}(608,1006) \put(0,0){\ing{acker2}} \put(281,988){\sx{3.}{$y$}} \put(279,895){\sx{3.}{$6$}} \put(279,795){\sx{3.}{$5$}} \put(279,694){\sx{3.}{$4$}} \put(279,594){\sx{3.}{$3$}} \put(279,468){\sx{3.}{$\mathrm e$}} \put(279,494){\sx{3.}{$2$}} \put(279,394){\sx{3.}{$1$}} \put(279,294){\sx{3.}{$0$}} \put(258,193){\sx{3.}{$-1$}} \put(258, 93){\sx{3.}{$-2$}} \put( 80,274){\sx{3.}{$-2$}} \put(180,274){\sx{3.}{$-1$}} \put(296,274){\sx{3.}{$0$}} \put(396,274){\sx{3.}{$1$}} \put(496,274){\sx{3.}{$2$}} \put(586,274){\sx{3.}{$x$}} \put(248,720){\sx{1.8}{\rot{85}$y\!=\!\mathcal{A}(4,x)\!=\! A_{2,4}(x\!+\!3)\!-\!3$\ero}} \put(312,720){\sx{1.8}{\rot{80}$y\!=\!\mathcal{A}(3,x)\!=\! A_{2,3}(x\!+\!3)\!-\!3$\ero}} \put(348,712){\sx{1.8}{\rot{63}$y\!=\!\mathcal{A}(2,x)\!=\! A_{2,2}(x\!+\!3)\!-\!3$\ero}} \put(314,526){\sx{1.8}{\rot{45}$y\!=\!\mathcal{A}(1,x)\!=\! A_{2,1}(x\!+\!3)\!-\!3$\ero}} \put(526,822){\sx{1.8}{\rot{82}$y\!=\!A_{2,4}(x)\!=\!\mathrm{tet}_2(x)$\ero}} \put(566,858){\sx{1.8}{\rot{73}$y\!=\!A_{2,3}(x)\!=\!2^x$\ero}} \put(566,792){\sx{1.8}{\rot{62}$y\!=\!A_{2,2}(x)\!=\!\mathrm{2}x$\ero}} \put(520,696){\sx{1.96}{\rot{44}$y\!=\!A_{2,1}(x)\!=\!2\!+\!x$\ero}} \put(32,326){\sx{1.9}{\rot{3}$y\!=\!2^x$\ero}} \put(132,4){\sx{1.9}{\rot{81}$y\!=\!\mathrm{tet}_2(x)$\ero}} \put(178,8){\sx{1.9}{\rot{66}$y\!=\!\mathrm{2} x$\ero}} \end{picture} \end{document}
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.
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 18:51, 11 March 2022 |  | 5,130 × 1,760 (1.92 MB) | Maintenance script (talk | contribs) | == Summary == Importing file |
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