Difference between revisions of "File:Superfactocomple1.png"

From TORI
Jump to navigation Jump to search
(Importing image file)
 
 
(2 intermediate revisions by one other user not shown)
Line 1: Line 1:
  +
{{oq|Superfactocomple1.png|Original file ‎(706 × 706 pixels, file size: 172 KB, MIME type: image/png)}}
−
Importing image file
  
  +
  +
[[Complex map]] of \(f=\)[[SuperFactorial]](\(x\!+\!\mathrm i y\)) in the \(x,y\) plane.
  +
  +
Levels <br>
  +
\(u\!=\!\Re(f)=\)constant and<br>
  +
\(u\!=\!\Im(f)=\)constant are drawn.<br>
  +
Thick lines correspond to the integer values.
  +
  +
==C++ generator of curves==
  +
Sorry, have misplaced the original generator. I load the code that does almost the same picture.
  +
  +
Files
  +
[[SuperFactorial.cin]]
  +
[[ado.cin]]
  +
[[conto.cin]]
  +
should be loaded to the working directory in order to compile the [[C++]] code below:
  +
<pre>
  +
#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 "fac.cin"
  +
//#include "sinc.cin"
  +
#include "facp.cin"
  +
#include "afacc.cin"
  +
#include "superfactorial.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("fig2b.eps","w");ado(o,402,402);
  +
FILE *o;o=fopen("SuperFacMap.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);
  +
// 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=-4;m<4;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(x,-8)L(x,8) fprintf(o,"0 setlinejoin 0 setlinecap 0.004 W 0 0 0 RGB S\n");
  +
M(x,0)L(-8.1,0) fprintf(o," .05 W 1 1 1 RGB S\n");
  +
DO(m,23){ M(x-.4*m,0)L(x-.4*(m+.5),0);} fprintf(o,".09 W .3 .3 0 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 SuperFacMap.eps");
  +
system( "open SuperFacMap.pdf"); //for LINUX
  +
// getchar(); system("killall Preview");//for mac
  +
}
  +
</pre>
  +
  +
== generator of labels==
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\paperwidth 342pt
  +
\paperheight 338pt
  +
\textwidth 500pt
  +
\textheight 500pt
  +
\topmargin -106pt
  +
\oddsidemargin -96pt
  +
\parindent 0pt
  +
\pagestyle{empty}
  +
\usepackage {graphics}
  +
\usepackage{rotating}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \sx {\scalebox}
  +
\begin{document}
  +
%\begin{picture}(1006,1006) \put(0,0){\ing{facit}}
  +
\begin{picture}(362,362)
  +
\put(0,0){\ing{SuperFacMap}}
  +
\put(30,357){\sx{1.3}{$y$}}
  +
\put(30,317){\sx{1.3}{$6$}}
  +
\put(30,277){\sx{1.3}{$4$}}
  +
\put(30,237){\sx{1.3}{$2$}}
  +
\put(29,196){\sx{1.3}{$0$}}
  +
\put(20,156){\sx{1.3}{$-2$}}
  +
\put(20,116){\sx{1.3}{$-4$}}
  +
\put(20,76){\sx{1.3}{$-6$}}
  +
\put(20,36){\sx{1.3}{$-8$}}
  +
\put(70,29){\sx{1.3}{$-6$}}
  +
\put(110,29){\sx{1.3}{$-4$}}
  +
\put(150,29){\sx{1.3}{$-2$}}
  +
\put(198,29){\sx{1.3}{$0$}}
  +
\put(238,29){\sx{1.3}{$2$}}
  +
\put(278,29){\sx{1.3}{$4$}}
  +
\put(318,29){\sx{1.3}{$6$}}
  +
\put(354,29){\sx{1.3}{$x$}}
  +
\put(50,344){\sx{1.3}{$u\!=\!2$}}
  +
\put(50,306){\sx{1.3}{$v\!=\!0$}}
  +
\put(50,255){\sx{1.3}{$u\!=\!2$}}
  +
\put(50,204){\sx{1.3}{$v\!=\!0$}} %central
  +
\put(50,152){\sx{1.3}{$u\!=\!2$}}
  +
\put(50,100){\sx{1.3}{$v\!=\!0$}}
  +
\put(50,049){\sx{1.3}{$u\!=\!2$}}
  +
% column<br>
  +
\put(122,342){\sx{1.2}{$v\!=\!-0.2$}}
  +
\put(135,314){\sx{1.2}{$u\!=\!1.8$}}
  +
\put(252,314){\sx{1.2}{$u\!=\!1.2$}}
  +
\put(136,265){\sx{1.2}{$v\!=\!0.2$}}
  +
\put(125,210){\sx{1.2}{$u\!=\!2.2$}}
  +
\put(125,130){\sx{1.2}{$v\!=\!-0.2$}}
  +
\put(134,084){\sx{1.2}{$u\!=\!1.8$}}
  +
\put(252,084){\sx{1.2}{$u\!=\!1.2$}}
  +
\put(134,054){\sx{1.2}{$v\!=\!0.2$}}
  +
% column<br>
  +
\put(322,343){\sx{1.3}{$u\!=\!1$}}
  +
\put(322,306){\sx{1.3}{$v\!=\!0$}}
  +
\put(322,269){\sx{1.3}{$u\!=\!1$}}
  +
\put(266,247){$u\!=\!0.8856031944$}
  +
\put(332,231){\sx{1.3}{$v\!=\!0$}} %central
  +
\put(329,164){\sx{1.3}{$v\!=\!0$}} %central
  +
\put(322,137){\sx{1.3}{$u\!=\!1$}}
  +
\put(322,100){\sx{1.3}{$v\!=\!0$}}
  +
\put(322, 50){\sx{1.3}{$u\!=\!1$}}
  +
\end{picture}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
==References==
  +
<references/>
  +
http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1<br>
  +
http://mizugadro.mydns.jp/PAPERS/2010superfae.pdf reprint, English version<br>
  +
http://mizugadro.mydns.jp/PAPERS/2010superfar.pdf reprint, Russian version<br>
  +
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)
  +
  +
==Keywords==
  +
«[[]]»,
  +
«[[SuperFactorial]]»,
  +
  +
[[Category:ComplexMap]]
  +
[[Category:ComplexMaps]]
  +
[[Category:SuperFactorial]]
  +
[[Category:Mathematical function]]
  +
[[Category:Mathematical functions]]
  +
[[Category:Superfunction]]
  +
[[Category:Superfunctions]]

Latest revision as of 00:10, 29 February 2024


Complex map of \(f=\)SuperFactorial(\(x\!+\!\mathrm i y\)) in the \(x,y\) plane.

Levels
\(u\!=\!\Re(f)=\)constant and
\(u\!=\!\Im(f)=\)constant are drawn.
Thick lines correspond to the integer values.

C++ generator of curves

Sorry, have misplaced the original generator. I load the code that does almost the same picture.

Files SuperFactorial.cin ado.cin conto.cin should be loaded to the working directory in order to compile 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 std::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 "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("fig2b.eps","w");ado(o,402,402);
 FILE *o;o=fopen("SuperFacMap.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);
 //     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=-4;m<4;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(x,-8)L(x,8) fprintf(o,"0 setlinejoin 0 setlinecap 0.004 W 0 0 0 RGB S\n");
 M(x,0)L(-8.1,0) fprintf(o,"                   .05 W  1 1 1 RGB S\n");
 DO(m,23){ M(x-.4*m,0)L(x-.4*(m+.5),0);} fprintf(o,".09 W  .3 .3 0 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 SuperFacMap.eps");     
        system(    "open SuperFacMap.pdf");     //for LINUX
 //     getchar(); system("killall Preview");//for mac
 }

generator of labels


\documentclass[12pt]{article}
\paperwidth 342pt
\paperheight 338pt
\textwidth 500pt
\textheight 500pt
\topmargin -106pt
\oddsidemargin -96pt
\parindent 0pt
\pagestyle{empty}
\usepackage {graphics}
\usepackage{rotating}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\begin{document}
%\begin{picture}(1006,1006) \put(0,0){\ing{facit}}
\begin{picture}(362,362)
\put(0,0){\ing{SuperFacMap}}
 \put(30,357){\sx{1.3}{$y$}}
 \put(30,317){\sx{1.3}{$6$}}
 \put(30,277){\sx{1.3}{$4$}}
 \put(30,237){\sx{1.3}{$2$}}
 \put(29,196){\sx{1.3}{$0$}}
 \put(20,156){\sx{1.3}{$-2$}}
 \put(20,116){\sx{1.3}{$-4$}}
 \put(20,76){\sx{1.3}{$-6$}}
 \put(20,36){\sx{1.3}{$-8$}}
 \put(70,29){\sx{1.3}{$-6$}}
 \put(110,29){\sx{1.3}{$-4$}}
 \put(150,29){\sx{1.3}{$-2$}}
 \put(198,29){\sx{1.3}{$0$}}
 \put(238,29){\sx{1.3}{$2$}}
 \put(278,29){\sx{1.3}{$4$}}
 \put(318,29){\sx{1.3}{$6$}}
 \put(354,29){\sx{1.3}{$x$}}
\put(50,344){\sx{1.3}{$u\!=\!2$}}
\put(50,306){\sx{1.3}{$v\!=\!0$}}
\put(50,255){\sx{1.3}{$u\!=\!2$}}
\put(50,204){\sx{1.3}{$v\!=\!0$}} %central
\put(50,152){\sx{1.3}{$u\!=\!2$}}
\put(50,100){\sx{1.3}{$v\!=\!0$}}
\put(50,049){\sx{1.3}{$u\!=\!2$}}
 % column<br>
\put(122,342){\sx{1.2}{$v\!=\!-0.2$}}
\put(135,314){\sx{1.2}{$u\!=\!1.8$}}
\put(252,314){\sx{1.2}{$u\!=\!1.2$}}
\put(136,265){\sx{1.2}{$v\!=\!0.2$}}
\put(125,210){\sx{1.2}{$u\!=\!2.2$}}
\put(125,130){\sx{1.2}{$v\!=\!-0.2$}}
\put(134,084){\sx{1.2}{$u\!=\!1.8$}}
\put(252,084){\sx{1.2}{$u\!=\!1.2$}}
\put(134,054){\sx{1.2}{$v\!=\!0.2$}}
% column<br>
\put(322,343){\sx{1.3}{$u\!=\!1$}}
\put(322,306){\sx{1.3}{$v\!=\!0$}}
\put(322,269){\sx{1.3}{$u\!=\!1$}}
\put(266,247){$u\!=\!0.8856031944$}
\put(332,231){\sx{1.3}{$v\!=\!0$}} %central
\put(329,164){\sx{1.3}{$v\!=\!0$}} %central
\put(322,137){\sx{1.3}{$u\!=\!1$}}
\put(322,100){\sx{1.3}{$v\!=\!0$}}
\put(322, 50){\sx{1.3}{$u\!=\!1$}}
\end{picture}
\end{document}

References

http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1
http://mizugadro.mydns.jp/PAPERS/2010superfae.pdf reprint, English version
http://mizugadro.mydns.jp/PAPERS/2010superfar.pdf reprint, 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)

Keywords

«[[]]», «SuperFactorial»,

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current17:50, 20 June 2013Thumbnail for version as of 17:50, 20 June 2013706 × 706 (172 KB)Maintenance script (talk | contribs)Importing image file

You cannot overwrite this file.

The following page uses this file:

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.

If the file has been modified from its original state, some details may not fully reflect the modified file.

Retrieved from "https://mizugadro.mydns.jp/t/index.php?title=File:Superfactocomple1.png&oldid=42793"