File:AbelFacMapT.png
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Complex map of function Abelfactorial (AuFac):
\[ u\!+\!\mathrm i v=\mathrm{AuFac}(x\!+\!\mathrm i y) \]
This function is described in the Moscow University Physics Bulletin, 2010 [1].
This map is used as Fig.8.7 at page 98 of book
«Superfunctions»[2][3]
in order to show the inverse function of Superfactorial «SuFac» is not complicated.
C++ Generator of curves
Files fac.cin, facp.cin, afacc.cin, superfactorial.cin, abelfac.cin, ado.cin and conto.cin should be loaded in the working directory for the compilation of the [[C++] 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 complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "fac.cin"
//#include "sinc.cin"
#include "facp.cin"
#include "afacc.cin"
#include "superfactorial.cin"
#include "abelfac.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=403,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("AbelFacMap.eps","w");ado(o,402,402);
fprintf(o,"201 201 translate\n 20 20 scale\n");
DO(m,M1)X[m]=-8.04+.04*(m+.5);
// DO(m,M1){t=-1.+.022*m; X[m]=.2+t-1.11*exp(-1.9*t);}
DO(n,N1)Y[n]=-8.04+.04*(n+.5);
// DO(n,N1){t=-8.04+.04*(n+.5); t*=.97; Y[n]=t-.25*sin(0.6127874523307*t);}
for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}}
for(n=-8;n<9;n++){ M( -8,n)L(8,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(m,M1){x=X[m]; //printf("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
// c=afacc(z);
// c=fac(z);
// c=superfac(z);
c=abelfac(z);
// p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.)-1.;
p=Re(c);q=Im(c);
if(p>-20 && p<20 &&
// (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&
q>-20 && q<20 && fabs(q)> 1.e-16
)
{g[m*N1+n]=p;f[m*N1+n]=q;}
}}
//fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1.8;q=.7;
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.4;q=.8;
for(m=-5;m<5;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".01 W 0 .5 0 RGB S\n");
for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".01 W .8 0 0 RGB S\n");
for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".01 W 0 0 .8 RGB S\n");
for(m=1;m<15;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".04 W .8 0 0 RGB S\n");
for(m=1;m<15;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 .8 RGB S\n");
conto(o,f,w,v,X,Y,M,N, (0. ),-9,9); fprintf(o,".04 W .5 0 .5 RGB S\n");
for(m=-14;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n");
m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-9,9); fprintf(o,".04 W 0 0 0 RGB S\n");
for(m=1;m<17;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n");
//#include"plofu.cin"
// x=0.8856031944;
// conto(o,g,w,v,X,Y,M,N,0.8856031944,-p,p); fprintf(o,".004 W .2 .2 0 RGB S\n");
M(2,0)L(-8.1,0) fprintf(o,"0 setlinejoin 0 setlinecap .05 W 1 1 1 RGB S\n");
DO(m,25){ M(2-.4*(m+.2),0)L(2-.4*(m+.4),0);} fprintf(o,".09 W 1 .4 0 RGB S\n");
DO(m,25){ M(2-.4*(m+.7),0)L(2-.4*(m+.9),0);} fprintf(o,".09 W 0 .4 1 RGB S\n");
//M(x,0)L(-8.1,0) fprintf(o,"[.19 .21]0 setdash .05 W 0 0 0 RGB S\n");
// May it be, that, some printers do not interpret well the dashing ?
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
system("epstopdf AbelFacMap.eps");
system( "open AbelFacMap.pdf"); //for LINUX
// getchar(); system("killall Preview");//for mac
}
Latex Generator of labels
% file AbelFacMap.pdf should be generated with the code above in order to compile the Latex document below.
% Gerenator of AbelFacMapT.png %<br>
% Copyleft 2012 by Dmitrii Kouznetsov %<br>
\documentclass[12pt]{article} %<br>
\usepackage{geometry} %<br>
\usepackage{graphicx} %<br>
\usepackage{rotating} %<br>
% \paperwidth 340pt %<br>
% \paperheight 336pt %<br>
\paperwidth 680pt %<br>
\paperheight 672pt %<br>
%\topmargin -96pt %<br>
%\oddsidemargin -98pt %<br>
\topmargin -104pt %<br>
\oddsidemargin -128pt %<br>
\textwidth 1100pt %<br>
\textheight 1100pt %<br>
\pagestyle {empty} %<br>
\newcommand \sx {\scalebox} %<br>
\newcommand \rot {\begin{rotate}} %<br>
\newcommand \ero {\end{rotate}} %<br>
\newcommand \ing {\includegraphics} %<br>
\parindent 0pt
\pagestyle{empty}
\begin{document}%<br>
\sx{2}{ \begin{picture}(362,362) %<br>
\put(0,0){\ing{AbelFacMap}} %<br>
\put(30,357){\sx{1.3}{$y$}} %<br>
\put(30,317){\sx{1.3}{$6$}} %<br>
\put(30,277){\sx{1.3}{$4$}} %<br>
\put(30,237){\sx{1.3}{$2$}} %<br>
\put(29,196){\sx{1.3}{$0$}} %<br>
\put(20,156){\sx{1.3}{$-2$}} %<br>
\put(20,116){\sx{1.3}{$-4$}} %<br>
\put(20,76){\sx{1.3}{$-6$}} %<br>
\put(20,36){\sx{1.3}{$-8$}} %<br>
\put(70,30){\sx{1.3}{$-6$}} %<br>
\put(110,30){\sx{1.3}{$-4$}} %<br>
\put(150,30){\sx{1.3}{$-2$}} %<br>
\put(198,30){\sx{1.3}{$0$}} %<br>
\put(238,30){\sx{1.3}{$2$}} %<br>
\put(278,30){\sx{1.3}{$4$}} %<br>
\put(318,30){\sx{1.3}{$6$}} %<br>
\put(354,30){\sx{1.3}{$x$}} %<br>
\put(50,306){\sx{1.3}{$u\!=\!1.8$}} %<br>
\put(50,255){\sx{1.3}{$v\!=\!0.6$}} %<br>
\put(90,198){\sx{1.3}{\bf cut}} %central%<br>
\put(50,150){\sx{1.3}{$v\!=\!-0.6$}} %<br>
\put(48,90){\sx{1.3}{$u\!=\!1.8$}} %<br>
% column<br>
\put(170,344){\sx{1.3}{$u\!=\!1.6$}} %<br>
%\put(250,314){\sx{1.3}{$u\!=\!1.2$}} %<br>
\put(119,230){\sx{1.3}{$v\!=\!0.8$}} %<br>
\put(109,170){\sx{1.3}{$v\!=\!-\!0.8$}} %<br>
\put(144,090){\sx{1.3}{$v\!=\!1.6$}} %<br>
\put(252,084){\sx{1.3}{$u\!=\!1.4$}} %<br>
% column<br>
\put(319,344){\sx{1.3}{$u\!=\!1.4$}} %<br>
\put(304,306){\sx{1.3}{$v\!=\!0.4$}} %<br>
\put(313,278){\sx{1.3}{$u\!=\!1.2$}} %<br>
\put(309,236){\sx{1.3}{$v\!=\!0.2$}} %<br>
%\put(266,247){\sx{1.}{$u\!=\!0.8856031944$}} %<br>
\put(306,209){\sx{1.3}{$u\!=\!1$}} %<br>
\put(331,198){\sx{1.3}{$v\!=\!0$}} %central%<br>
\put(312,118){\sx{1.3}{$u\!=\!1.2$}} %<br>
\put(304,90){\sx{1.3}{$v\!=\!-0.4$}} %<br>
\put(320, 50){\sx{1.3}{$u\!=\!1.4$}} %<br>
\end{picture} %<br>
} %<br>
\end{document} %<br>
%
References
- ↑
http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1
http://mizugadro.mydns.jp/PAPERS/2010superfae.pdf English version
http://mizugadro.mydns.jp/PAPERS/2010superfar.pdf Russian version
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12. (Russian version: p.8-14) - ↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics - Lambert Academic Publishing, 2020/7/28
- ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.
Keywords
«Abelfactorial», «Abelfunction», «AuFac», «Factorial», «Iterate», «Regular iteration», «SuFac», «Superfactorial», «Superfunction», «Superfunctions», «Square root of factorial»,
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 17:50, 20 June 2013 | | 1,412 × 1,395 (372 KB) | Maintenance script (talk | contribs) | Importing image file |
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