Difference between revisions of "File:LambertWmap150.png"

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{{oq|LambertWmap150.png|Original file ‎(1,773 × 1,752 pixels, file size: 523 KB, MIME type: image/png) }}
−
Importing image file
  
  +
[[Complex map]] of the [[LambertW function]].
  +
  +
\(f=\mathrm{LambertW}(x\!+\!\mathrm i y)\) is shown with levels
  +
  +
\(\Re(f)=u=\mathrm{const~}\) and
  +
\(\Im(f)=v=\mathrm{const~}\).
  +
  +
The representation through the [[Tania function]] is used.
  +
  +
This map is used as Fig.11.4 at page 138 of book «[[Superfunctions]]», 2020
  +
<ref>
  +
https://www.amazon.com/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862
  +
Dmitrii Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  +
Tools for evaluation of superfunctions, abelfunctions and non-integer iterates of holomorphic functions are collected. For a given transferfunction T, the superfunction is solution F of the transfer equation F(z+1)=T(F(z)) . The abelfunction is inverse of F. In particular, superfunctions of factorial, exp, sin are suggested. The Holomorphic extensions of the logistic sequence and those of the Ackermann functions are considered. Among ackermanns, the tetration (mainly to the base b>1) and natural pentation (to base b=e) are presented. The efficient algorithm for the evaluation of superfunctions and abelfunctions are described. The graphics and complex maps are plotted. The possible applications are discussed. Superfunctions significantly extend the set of functions, that can be used in scientific research and technical design.
  +
</ref><ref>
  +
https://mizugadro.mydns.jp/BOOK/468.pdf
  +
Dmitrii Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  +
</ref><br>
  +
in order to show the inverse function of the [[Transfer function]] considered in Chapter 11 as an example for the [[exotic iteration]].
  +
  +
==Generators==
  +
  +
===[[C++]] generator of curves===
  +
  +
Files [[ado.cin]] and [[conto.cin]] should be loaded to the working directory in order to compile the 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 complex<double> z_type;
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
#include "conto.cin"
  +
  +
z_type ArcTania(z_type z) {return z + log(z) - 1. ;}
  +
z_type ArcTaniap(z_type z) {return 1. + 1./z ;}
  +
  +
z_type TaniaTay(z_type z) { int n; z_type s;
  +
s=1.+z*(.5+z*(1./16.+z*(-1./192.+z*(-1./3072.+z*(1.3/6144.+z*(-4.7/147456.
  +
//+z*(7.3/4128768.) //some reserve term
  +
)))))); DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
  +
  +
z_type TaniaNega(z_type z){int n;z_type s=exp(z-exp(z)+1.);
  +
DO(n,4) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
  +
  +
z_type TaniaBig(z_type z){int n;z_type s=z; s=z-log(s)+1.;
  +
DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
  +
  +
z_type TaniaS(z_type z){int n; z_type s,t=z+z_type(2.,-M_PI);t*=2./9.; t=I*sqrt(t);
  +
s=-1.+t*(3.+t*(-3.+t*(.75+t*(.3+t*(.9/16.+t*(-.3/7.+t*(-12.51/224. //+t*(-.9/28.)
  +
)))))));
  +
DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
  +
  +
z_type Tania(z_type z){ z_type t;
  +
if( fabs(Im(z))< M_PI && Re(z)<-2.51) return TaniaNega(z);
  +
if( abs(z)>7. || Re(z)>3.8 ) return TaniaBig(z);
  +
if( Im(z) > .7 ) return TaniaS(z);
  +
if( Im(z) < -.7) return conj(TaniaS(conj(z)));
  +
return TaniaTay(z);
  +
}
  +
  +
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
int M=160,M1=M+1;
  +
int N=161,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("productlogmap.eps","w");ado(o,162,162);
  +
fprintf(o,"81 81 translate\n 10 10 scale\n");
  +
DO(m,M1) X[m]=-8.+.1*(m);
  +
DO(n,80)Y[n]=-8.+.1*n;
  +
Y[80]=-.033;
  +
Y[81]= .033;
  +
for(n=82;n<N1;n++) Y[n]=-8.+.1*(n-1.);
  +
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=Tania(log(z)-1.); 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;}
  +
}}
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=.6;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,".01 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,".01 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,".01 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,".05 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,".05 W 0 0 .9 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".05 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,".05 W 0 0 0 RGB S\n");
  +
y=0.; for(m=0;m<75;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)}
  +
fprintf(o,".07 W 1 .5 0 RGB S\n");
  +
y=0.; for(m=2;m<75;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)}
  +
fprintf(o,".07 W 0 .5 1 RGB S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf productlogmap.eps");
  +
system( "open productlogmap.pdf");
  +
getchar(); system("killall Preview");
  +
}
  +
</pre>
  +
===[[Latex]] generator of lables===
  +
<pre>
  +
% Copyleft 2011 by Dmitrii Kouznetsov %<br>
  +
\documentclass[12pt]{article} %<br>
  +
\usepackage{geometry} %<br>
  +
\usepackage{graphicx} %<br>
  +
\usepackage{rotating} %<br>
  +
\paperwidth 854pt %<br>
  +
\paperheight 844pt %<br>
  +
\topmargin -96pt %<br>
  +
\oddsidemargin -98pt %<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>
  +
\begin{document} %<br>
  +
\sx{5}{ \begin{picture}(164,165) %<br>
  +
%\put(6,5){\ing{taniacontour}} %<br>
  +
\put(6,5){\ing{productlogmap}} %<br>
  +
\put(2,162){\sx{.7}{$y$}} %<br>
  +
\put(2,144){\sx{.6}{$6$}} %<br>
  +
\put(2,124){\sx{.6}{$4$}} %<br>
  +
\put(2,104){\sx{.6}{$2$}} %<br>
  +
\put(4,118){ \sx{.8}{\rot{-36}$v\!=\!1.8$\ero}} %<br>
  +
\put(3,96){ \sx{.8}{\rot{-20}$v\!=\!2$\ero}} %<br>
  +
\put(2, 84){\sx{.6}{$0$}} %<br>
  +
\put(8, 84){\sx{.8}{\bf cut}} %<br>
  +
\put(7,72){\sx{.7}{\rot{18}$v\!=\!-\!2$\ero}} %<br>
  +
\put(-3,64){\sx{.6}{$-2$}} %<br>
  +
\put(-3,44){\sx{.6}{$-4$}} %<br>
  +
\put(-3,24){\sx{.6}{$-6$}} %<br>
  +
\put( 22,0){\sx{.6}{$-6$}} %<br>
  +
\put( 42,0){\sx{.6}{$-4$}} %<br>
  +
\put( 62,0){\sx{.6}{$-2$}} %<br>
  +
\put( 86,0){\sx{.6}{$0$}} %<br>
  +
\put(106,0){\sx{.6}{$2$}} %<br>
  +
\put(126,0){\sx{.6}{$4$}} %<br>
  +
\put(146,0){\sx{.6}{$6$}} %<br>
  +
\put(164,0){\sx{.7}{$x$}} %<br>
  +
\put( 41, 76){\rot{-39}\sx{.8}{$u\!=\!0.8$}\ero}%<br>
  +
\put( 41, 57){\rot{-26}\sx{.84}{$u\!=\!1$}\ero}%<br>
  +
\put( 39, 40){\rot{-19}\sx{.8}{$u\!=\!1.2$}\ero}%<br>
  +
\put( 33, 21){\rot{-18}\sx{.8}{$u\!=\!1.4$}\ero}%<br>
  +
\put( 85, 145){\rot{83}\sx{.82}{$v\!=\!1$}\ero}%<br>
  +
\put(137, 102){\rot{17}\sx{.8}{$v\!=\!0.2$}\ero}%<br>
  +
\put(144, 84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}%<br>
  +
\put(140, 65){\rot{-16}\sx{.72}{$v\!=\!-\!0.2$}\ero}%<br>
  +
\put(134, 46){\rot{-32}\sx{.72}{$v\!=\!-0.4$}\ero}%<br>
  +
\put( 79, 33){\rot{-80}\sx{.86}{$v\!=\!-1$}\ero}%<br>
  +
\end{picture} %<br>
  +
} %<br>
  +
\end{document}
  +
</pre>
  +
==References==
  +
{{ref}}
  +
  +
http://mathworld.wolfram.com/LambertW-Function.html
  +
  +
{{fer}}
  +
==Keywords==
  +
  +
«[[ArcLambertW]]»,
  +
«[[ArcZex]]»,
  +
«[[Exotic iteration]]»,
  +
«[[Exotic iteration]]»,
  +
«[[Fixed point]]»,
  +
«[[LambertW]]»,
  +
«[[ProductLog]]»,
  +
«[[Superfunctions]]»,
  +
«[[Transfer function]]»,
  +
«[[Zex]]»,
  +
  +
«[[Суперфункции]]»,
  +
  +
[[Category:ArcZex]]
  +
[[Category:Book]]
  +
[[Category:BookE]]
  +
[[Category:BookMap]]
  +
[[Category:BookMapE]]
  +
[[Category:C++]]
  +
[[Category:Complex map]]
  +
[[Category:Exotic iteration]]
  +
[[Category:Inverse functions]]
  +
[[Category:Iterate]]
  +
[[Category:LambertW]]
  +
[[Category:LambertW function]]
  +
[[Category:Laser science]]
  +
[[Category:Latex]]
  +
[[Category:Mahtematics of Computation]]
  +
[[Category:Special function]]
  +
[[Category:Superfunction]]
  +
[[Category:Superfunctions]]
  +
[[Category:Tania function]]
  +
[[Category:Transfer function]]
  +
[[Category:zex]]

Latest revision as of 15:16, 26 August 2025


Complex map of the LambertW function.

\(f=\mathrm{LambertW}(x\!+\!\mathrm i y)\) is shown with levels

\(\Re(f)=u=\mathrm{const~}\) and \(\Im(f)=v=\mathrm{const~}\).

The representation through the Tania function is used.

This map is used as Fig.11.4 at page 138 of book «Superfunctions», 2020 [1][2]
in order to show the inverse function of the Transfer function considered in Chapter 11 as an example for the exotic iteration.

Generators

C++ generator of curves

Files ado.cin and conto.cin 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++)
 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 "conto.cin"

 z_type ArcTania(z_type z) {return z + log(z) - 1. ;}
 z_type ArcTaniap(z_type z) {return 1. + 1./z ;}

 z_type TaniaTay(z_type z) { int n; z_type s;
 s=1.+z*(.5+z*(1./16.+z*(-1./192.+z*(-1./3072.+z*(1.3/6144.+z*(-4.7/147456.
 //+z*(7.3/4128768.) //some reserve term
 )))))); DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }

 z_type TaniaNega(z_type z){int n;z_type s=exp(z-exp(z)+1.); 
 DO(n,4) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }

 z_type TaniaBig(z_type z){int n;z_type s=z; s=z-log(s)+1.; 
 DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }

 z_type TaniaS(z_type z){int n; z_type s,t=z+z_type(2.,-M_PI);t*=2./9.; t=I*sqrt(t);
 s=-1.+t*(3.+t*(-3.+t*(.75+t*(.3+t*(.9/16.+t*(-.3/7.+t*(-12.51/224. //+t*(-.9/28.)
 )))))));
 DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }

 z_type Tania(z_type z){ z_type t;
 if( fabs(Im(z))< M_PI && Re(z)<-2.51) return TaniaNega(z);
 if( abs(z)>7. || Re(z)>3.8 ) return TaniaBig(z);
 if( Im(z) > .7 ) return TaniaS(z);
 if( Im(z) < -.7) return conj(TaniaS(conj(z)));
 return TaniaTay(z);
 }

 main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  int M=160,M1=M+1;
  int N=161,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("productlogmap.eps","w");ado(o,162,162);
 fprintf(o,"81 81 translate\n 10 10 scale\n");
 DO(m,M1) X[m]=-8.+.1*(m);
 DO(n,80)Y[n]=-8.+.1*n;
         Y[80]=-.033;
         Y[81]= .033;
 for(n=82;n<N1;n++) Y[n]=-8.+.1*(n-1.);
 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=Tania(log(z)-1.); 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;}
         }}
 fprintf(o,"1 setlinejoin 2 setlinecap\n");  p=.6;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,".01 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,".01 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,".01 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,".05 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,".05 W 0 0 .9 RGB S\n");
                    conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".05 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,".05 W 0 0 0 RGB S\n");
 y=0.; for(m=0;m<75;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)}
 fprintf(o,".07 W 1 .5 0 RGB S\n");
 y=0.; for(m=2;m<75;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)}
 fprintf(o,".07 W 0 .5 1 RGB S\n");
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf productlogmap.eps");   
        system(    "open productlogmap.pdf");
        getchar(); system("killall Preview");
 }

Latex generator of lables

 % Copyleft 2011 by Dmitrii Kouznetsov %<br>
 \documentclass[12pt]{article} %<br>
 \usepackage{geometry} %<br>
 \usepackage{graphicx} %<br>
 \usepackage{rotating} %<br>
 \paperwidth 854pt %<br>
 \paperheight 844pt %<br>
 \topmargin -96pt %<br>
 \oddsidemargin -98pt %<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>
 \begin{document} %<br>
 \sx{5}{ \begin{picture}(164,165) %<br>
 %\put(6,5){\ing{taniacontour}} %<br>
 \put(6,5){\ing{productlogmap}} %<br>
 \put(2,162){\sx{.7}{$y$}} %<br>
 \put(2,144){\sx{.6}{$6$}} %<br>
 \put(2,124){\sx{.6}{$4$}} %<br>
 \put(2,104){\sx{.6}{$2$}} %<br>
 \put(4,118){ \sx{.8}{\rot{-36}$v\!=\!1.8$\ero}} %<br>
 \put(3,96){ \sx{.8}{\rot{-20}$v\!=\!2$\ero}} %<br>
 \put(2, 84){\sx{.6}{$0$}} %<br>
 \put(8, 84){\sx{.8}{\bf cut}} %<br>
 \put(7,72){\sx{.7}{\rot{18}$v\!=\!-\!2$\ero}} %<br>
 \put(-3,64){\sx{.6}{$-2$}} %<br>
 \put(-3,44){\sx{.6}{$-4$}} %<br>
 \put(-3,24){\sx{.6}{$-6$}} %<br>
 \put( 22,0){\sx{.6}{$-6$}} %<br>
 \put( 42,0){\sx{.6}{$-4$}} %<br>
 \put( 62,0){\sx{.6}{$-2$}} %<br>
 \put( 86,0){\sx{.6}{$0$}} %<br>
 \put(106,0){\sx{.6}{$2$}} %<br>
 \put(126,0){\sx{.6}{$4$}} %<br>
 \put(146,0){\sx{.6}{$6$}} %<br>
 \put(164,0){\sx{.7}{$x$}} %<br>
 \put( 41, 76){\rot{-39}\sx{.8}{$u\!=\!0.8$}\ero}%<br>
 \put( 41, 57){\rot{-26}\sx{.84}{$u\!=\!1$}\ero}%<br>
 \put( 39, 40){\rot{-19}\sx{.8}{$u\!=\!1.2$}\ero}%<br>
 \put( 33, 21){\rot{-18}\sx{.8}{$u\!=\!1.4$}\ero}%<br>
 \put( 85, 145){\rot{83}\sx{.82}{$v\!=\!1$}\ero}%<br>
 \put(137, 102){\rot{17}\sx{.8}{$v\!=\!0.2$}\ero}%<br>
 \put(144,  84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}%<br>
 \put(140,  65){\rot{-16}\sx{.72}{$v\!=\!-\!0.2$}\ero}%<br>
 \put(134,  46){\rot{-32}\sx{.72}{$v\!=\!-0.4$}\ero}%<br>
 \put( 79,  33){\rot{-80}\sx{.86}{$v\!=\!-1$}\ero}%<br>
 \end{picture} %<br>
 } %<br>
 \end{document}

References

  1. ↑ https://www.amazon.com/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020. Tools for evaluation of superfunctions, abelfunctions and non-integer iterates of holomorphic functions are collected. For a given transferfunction T, the superfunction is solution F of the transfer equation F(z+1)=T(F(z)) . The abelfunction is inverse of F. In particular, superfunctions of factorial, exp, sin are suggested. The Holomorphic extensions of the logistic sequence and those of the Ackermann functions are considered. Among ackermanns, the tetration (mainly to the base b>1) and natural pentation (to base b=e) are presented. The efficient algorithm for the evaluation of superfunctions and abelfunctions are described. The graphics and complex maps are plotted. The possible applications are discussed. Superfunctions significantly extend the set of functions, that can be used in scientific research and technical design.
  2. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.

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