Difference between revisions of "File:SuZexD1mapT.png"

Latest revision as of 16:04, 26 August 2025

It is entire superfunction of Zex; \[ \mathrm{zex}(z)=z\exp(z) \] \[ u\!+\!\mathrm i v = \mathrm{SuZex}(c\!+\!\mathrm i y)\]

SuZex is built-up at the fixed point zero from asymptotic behavior; the approximation below is implemented:

\(\!\! (1) ~ ~ ~ \displaystyle f(z)= \frac{-1}{z} +\frac{ \ln(-z)}{2 z^2}+ \frac{-.05\ln(-z)^2-.02\ln(-z)-.4}{z^3} \)

For integer \(n\),

\(\!\! (2) ~ ~ ~ \displaystyle F_n(z)=\mathrm{zex}^n\Big(f(z\!-\!n) \Big)\)

The constant \(x_n\) is chosen as solution of equation \(F_n(x_n)\!=\!1\). Then, the superfunction is evaluated as follows:

\(\!\! (2) ~ ~ ~ \displaystyle \mathrm{SuZex}(z)=F_n(x_n\!+\!z)\)

for integer \(n\).

The generator below uses value \(n\!=\!16\); it is sufficient to get the camera-ready copy. For the precise computation, more terms in the expansion (1) should be calculated. The precise implementation (with 14 decimal digits) is loaded as SuZex.cin.

The map is used as Fig.11.6 at page 143 of book «Superfunctions», 2020 ^[1]^[2]
in order to show the fast growth of the function along the real axis.

C++ generator of curves

 #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"
 // #include "fsexp.cin"
 //#include "fslog.cin"

 z_type zex(z_type z){ return z*exp(z);} 

 int Nitera=16;

 z_type suzex0b(z_type z){ return -1./z; }
// z_type suzex0(z_type z){ z_type L=log(-z); return (-1.+.5*L/z)/z; }
 z_type suzex0(z_type z){ z_type L=log(-z); return (-1.+(.5*L + (-.05*L*L-.02*L-.4)/z)/z)/z; }

 z_type suzexn(int n, z_type z){int m;  z-=0.+n; z=suzex0(z); DO(m,n) z=zex(z); return z; } 

 main(){ int j,k,m,n; DB x1,x,y, p,q, t; z_type z,c,d, cu,cd;
 x1=-1.04;
 DO(n,18){ y=Re(suzexn(Nitera,x1)); x=y-1.; x1-=1.5*x; printf("%18.16f %18.16f\n", x1,y);}
 getchar();

 int M=601,M1=M+1;
 int N=401,N1=N+1;
 DB X[M1],Y[N1];
 DB *g, *f, *w; // w is working array.
 g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
 f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
 w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
 char v[M1*N1]; // v is working array
 FILE *o;o=fopen("suZexMap.eps","w");  ado(o,1202,1202);
 fprintf(o,"601 601 translate\n 100 100 scale\n");
 fprintf(o,"1 setlinejoin 2 setlinecap\n");
 DO(m,M1) X[m]=-6.+.02*(m-.1);
 //DO(n,N1) Y[n]=-5.+.02*(n-.5); 
 for(n=0;n<N1;n++) { Y[n]=0.6*sinh((3./200.)*(n-200.5)); printf("%3d %9.6f\n",n,Y[n]); }
 for(m=-6;m<7;m++) {M(m,-6)L(m,6)}
 for(n=-6;n<7;n++) {M(  -6,n)L(6,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
 DO(m,M1)DO(n,N1){      g[m*N1+n]=999;
                        f[m*N1+n]=999;}
 DO(m,M1){x=X[m]; if(m/10*10==m) printf("x=%6.3f\n",x);
 DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.019)
       { 
 c=suzexn(Nitera,z+x1);
         p=Re(c); q=Im(c);
         if(p>-19 && p<19 &&  fabs(q)>1.e-12 && fabs(p)>1.e-12) g[m*N1+n]=p;
         if(p>-19 && p<19 &&  fabs(q)>1.e-12 && fabs(p)>1.e-12) f[m*N1+n]=q;
       }
        }}
 fprintf(o,"1 setlinejoin 1 setlinecap\n");
 p=2.;q=.3;

 for(m=-8;m<8;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".007 W 0 .6 0 RGB S\n");
 for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".007 W .9 0 0 RGB S\n");
 for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".007 W 0 0 .9 RGB S\n");
 for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".02 W .8 0 0 RGB S\n");
 for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".02 W 0 0 .8 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".02 W .5 0 .5 RGB S\n");
 for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");

 //#include "plofu.cin"
 fprintf(o,"0 setlinejoin 0 setlinecap\n");
 fprintf(o,"showpage\n");
 fprintf(o,"%c%cTrailer\n",'%','%');
 fclose(o);  free(f); free(g); free(w);
       system("epstopdf SuZexMap.eps"); 
       system(    "open SuZexMap.pdf"); //for macintosh
       getchar(); system("killall Preview"); // For macintosh
 }

Latex generator of labels

%

\documentclass[12pt]{article} % <br>
\paperheight 1228px %  <br>
\paperwidth 1236px %  <br>
\textwidth 1394px %  <br>
\textheight 1300px %  <br>
\topmargin -104px %  <br>
\oddsidemargin -78px %  <br>
\usepackage{graphics} %  <br>
\usepackage{rotating} %  <br>
\newcommand \sx {\scalebox} %  <br>
\newcommand \rot {\begin{rotate}} %  <br>
\newcommand \ero {\end{rotate}} %  <br>
\newcommand \ing {\includegraphics} %  <br>
\newcommand \rmi {\mathrm{i}} %  <br>
\begin{document} %  <br>
\newcommand \zoomax { %  <br>
\put(18,1206){\sx{3.3}{$y$}} %  <br>
\put(18,1113){\sx{3}{$5$}} %  <br>
\put(18,1013){\sx{3}{$4$}} %  <br>
\put(18, 913){\sx{3}{$3$}} %  <br>
\put(18, 813){\sx{3}{$2$}} %  <br>
\put(18, 713){\sx{3}{$1$}} %  <br>
\put(18, 613){\sx{3}{$0$}} %  <br>
\put(-6, 513){\sx{3}{$-1$}} %  <br>
\put(-6, 413){\sx{3}{$-2$}} %  <br>
\put(-6, 313){\sx{3}{$-3$}} %  <br>
\put(-6, 213){\sx{3}{$-4$}} %  <br>
\put(-6, 113){\sx{3}{$-5$}} %  <br>
\put(-6, 013){\sx{3}{$-6$}} %  <br>
\put(014, -5){\sx{3}{$-6$}} %  <br>
\put(114, -5){\sx{3}{$-5$}} %  <br>
\put(214, -5){\sx{3}{$-4$}} %  <br>
\put(314, -5){\sx{3}{$-3$}} %  <br>
\put(414, -5){\sx{3}{$-2$}} %  <br>
\put(514, -5){\sx{3}{$-1$}} %  <br>
\put(635, -5){\sx{3}{$0$}} %  <br>
\put(735, -5){\sx{3}{$1$}} %  <br>
\put(835, -5){\sx{3}{$2$}} %  <br>
\put(935, -5){\sx{3}{$3$}} %  <br>
\put(1035, -5){\sx{3}{$4$}} %  <br>
\put(1135, -5){\sx{3}{$5$}} %  <br>
\put(1227,-4){\sx{3}{$x$}} %  <br>
} %  <br>
\parindent 0pt %  <br>
\sx{1}{\begin{picture}(1252,1220) % <br>
%\put(40,20){\ing{b271tMap3}} %  <br>
%\put(40,20){\ing{ExpMap}} %  <br>
\put(40,20){\ing{SuZexMap}} %  <br>
\zoomax %  <br>
\put(290,611){\sx{4}{$v\!=\!0$}} %  <br>
\put(183,560){\sx{4}{\rot{90}$u\!=\!0.2$\ero}} %  <br>
\put(468,560){\sx{4}{\rot{90}$u\!=\!0.4$\ero}} %  <br>
\put(696,118){\sx{4}{\rot{83}$u\!=\!0$\ero}} %<br>
\put(980,236){\sx{4}{\rot{24}$u\!=\!-0.2$\ero}} %<br>
\put(790, 44){\sx{4}{\rot{38}$v\!=\!-0.2$\ero}} %<br>
\end{picture}} %  <br>
\end{document} %  <br>

Warning

The map had been generated before the expansion (11.12) at page 142 of book ^[1]^[2] had been deduced; the last term of the fit used does not coincide with the 3d term of the asymptotic expansion.

Colleagues are invited to generate this map using the truncated asymptotic expansion and to check that the resulting picture is the same.

For plotting of the compex map, it happens to be easier and faster to iterate zex few times more than to calculate more terms in the expansion.

References

↑ ^1.0 ^1.1 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.
↑ ^2.0 ^2.1 https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.

http://mathworld.wolfram.com/LambertW-Function.html

Keywords

«ArcLambertW», «ArcZex», «Exotic iteration», «Fixed point», «LambertW», «ProductLog», «Superfunction», «Superfunctions», «SuZex», «Zex»,

«Суперфункции»,

File history

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	Date/Time	Thumbnail	Dimensions	User	Comment
current	17:50, 20 June 2013		2,576 × 2,559 (1.04 MB)	Maintenance script (talk \| contribs)	Importing image file

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File usage

The following page uses this file:

SuZex

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.

Horizontal resolution	59.06 dpc
Vertical resolution	59.06 dpc

[a-1] 1.0 ^1.1 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.

[t-2] 2.0 ^2.1 https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.

[1]

[2]

@@ Line 1: / Line 1: @@
+{{oq|SuZexD1mapT.png|Original file ‎(2,576 × 2,559 pixels, file size: 1.04 MB, MIME type: image/png) }}
-[[Complex map]] of function [[SuZex]], which is entire [[superfunction]] of [[Zex]], $\mathrm{zex}(z)=z\exp(z)$.
+[[Complex map]] of function [[SuZex]].
+It is entire [[superfunction]] of [[Zex]];
-$u\!+\!\mathrm i v = \mathrm{SuZex}(c\!+\!\mathrm i y)$
+\[
+\mathrm{zex}(z)=z\exp(z)
+\]
+\[
+u\!+\!\mathrm i v = \mathrm{SuZex}(c\!+\!\mathrm i y)\]
 [[SuZex]] is built-up at the fixed point zero from asymptotic behavior; the approximation below is implemented:
-:$\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ \displaystyle
+:\(\!\! (1) ~ ~ ~ \displaystyle
-f(z)= \frac{-1}{z} +\frac{ \ln(-z)}{2 z^2}+ \frac{-.05\ln(-z)^2-.02\ln(-z)-.4}{z^3}$
+f(z)= \frac{-1}{z} +\frac{ \ln(-z)}{2 z^2}+ \frac{-.05\ln(-z)^2-.02\ln(-z)-.4}{z^3}
+\)
-For integer $n$,
-:$\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \displaystyle
-F_n(z)=\mathrm{zex}^n\Big( z\!-\!n) \Big)$
-The constant $x_n$ is chosen as solution of equation $F_n(x_n)\!=\!1$. Then, the superfunction is evaluated as iteration
-:$\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \displaystyle
-\mathrm{SuZex}(z)=F_n(x_n\!+\!z)$
-for integer $n$.
+For integer \(n\),
-The generator below uses value $n\!=\!16$, which is sufficient to get the camera-ready copy.  For the precise computation, more terms in the expansion (1) should be calculated.
+:\(\!\! (2) ~ ~ ~ \displaystyle
+F_n(z)=\mathrm{zex}^n\Big(f(z\!-\!n) \Big)\)
+The constant \(x_n\) is chosen as solution of equation \(F_n(x_n)\!=\!1\). Then, the superfunction is evaluated as follows:
+:\(\!\! (2) ~ ~ ~ \displaystyle
+\mathrm{SuZex}(z)=F_n(x_n\!+\!z)\)
+for integer \(n\).
+The generator below uses value \(n\!=\!16\); it is sufficient to get the camera-ready copy.  For the precise computation, more terms in the expansion (1) should be calculated.
 The precise implementation (with 14 decimal digits) is loaded as [[SuZex.cin]].
+The map is used as Fig.11.6 at page 143 of book «[[Superfunctions]]», 2020
+<ref name="a">
+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 name="t">
+https://mizugadro.mydns.jp/BOOK/468.pdf
+Dmitrii Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
+</ref><br>
+in order to show the fast growth of the function along the real axis.
 ==[[C++]] generator of curves==
+<pre>
  #include <math.h>
  #include <stdio.h>
@@ Line 95: / Line 116: @@
        getchar(); system("killall Preview"); // For macintosh
  }
+</pre>
 ==[[Latex]] generator of labels==
+% <pre>
-% <nowiki> %<br>
 \documentclass[12pt]{article} % <br>
 \paperheight 1228px %  <br>
@@ Line 157: / Line 177: @@
 \end{picture}} %  <br>
 \end{document} %  <br>
-</nowiki>
+</pre>
+==Warning==
+The map had been generated before the expansion (11.12) at page 142 of book <ref name="a"/><ref name="t"/>
+had been deduced; the last term of the fit used does not coincide with the 3d term
+of the asymptotic expansion.
+Colleagues are invited to generate this map using the truncated asymptotic expansion
+and to check that the resulting picture is the same.
+For plotting of the compex map, it happens to be easier and faster to iterate zex few times more than to calculate more terms in the expansion.
+==References==
+{{ref}}
+http://mathworld.wolfram.com/LambertW-Function.html
+{{fer}}
+==Keywords==
+«[[ArcLambertW]]»,
+«[[ArcZex]]»,
+«[[Exotic iteration]]»,
+«[[Fixed point]]»,
+«[[LambertW]]»,
+«[[ProductLog]]»,
+«[[Superfunction]]»,
+«[[Superfunctions]]»,
+«[[SuZex]]»,
+«[[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:SuZex]]
-[[Category:Zex]]
 [[Category:Superfunction]]
-[[Category:Regular iteration]]
+[[Category:Superfunctions]]
+[[Category:Tania function]]
+[[Category:Transfer function]]
+[[Category:zex]]