Difference between revisions of "File:Analuxp02t900.jpg"

Latest revision as of 01:20, 16 December 2025

Figure 14.5 from page 187 of book «Superfunctions» ^[1], 2020.
Similar maps appear also in Рис.14.5 at page 187 of the Russian version «Суперфункции»^[2], 2014.
Even earlier, these maps are published in Mathematics of Computation ^[3], 2009.

The figure shows the Map of function

$ \displaystyle K_A(z)= L\left(\frac{1}{2}-\frac{1}{2\pi\mathrm i}\ln\frac{1-\mathrm i A+z}{ 1+\mathrm i A-z} \right) ~+~L^*\left(\frac{1}{2}-\frac{1}{2\pi\mathrm i}\ln\frac{1-\mathrm i A-z}{ 1+\mathrm i A+z} \right) $

where $L=~$filog$(1)\approx .31813150520476413 + 1.3372357014306895 \, \mathrm i$

is fixed point of natural logarithm, $\ln(L)\!=\!L\!=\!\exp(L)$. This is asymptotic value of natural tetration in the Second quadrant of the complex plane.

The maps are shown with levels of logamplitude $\rho$ and phase $\varphi$;

$ \exp(\rho \!+\! \mathrm i \varphi)=K_A(x\!+\!\mathrm i y)$

The maps are plotted for $A=3$, $A=5$, and $A=10$.

This is analogy of figure 3 by ^[3].

The shaded strip $|x|\!\le\! 0.5$, $|y| \!\le\! 4.5$ indicates the range of values, requested for evaluation of tetration in the last picture of figure http://mizugadro.mydns.jp/t/index.php/File:Analuxp01u400.jpg In the shaded region, the phase of function $K_A$ does not approach $\pi$; apparently, there $|\varphi|<2$.

C++ generator of curves for the first map

// Some operational systems do not recognize the extensions,
// they confuse the filename of the picture with filename of its generator.
// Therefore, the filename of generator of a picture has an assitional "z" at the end of name.
#include<complex.h>
#define Re(z) z.real()
#define Im(z) z.imag()
#define z_type complex<double>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
//#include "ado.cin"
//#include"advacon.cin"
#include"conto.cin"
//#include"f3c.cin"
#define Zo z_type(.31813150520476413, 1.3372357014306895)
#define Zc z_type(.31813150520476413,-1.3372357014306895)
#define I z_type(0.,1.)
int main(){ int m,n,j; DB x,y, A,Ay; z_type c,z;

int M= 81,M1=M+1;
int N=101,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
printf("Output analuxp02a.eps\n");
FILE *o;o=fopen("analuxp02a.eps","w");ado(o,82,104);
fprintf(o,"41 52 translate\n 10 10 scale\n");
//#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
//#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);

//A=3; M(0,-A-.05)L(0,A+.05)fprintf(o,"1 .8 1 RGB 1 W S\n");
A=3; M(0,-4.5)L(0,4.5)fprintf(o,".8 1 .8 RGB 1 W S\n");
//DB sy=3./sinh(.1*N/2.);
DO(m,M1) X[m]=.1*(m-M/2-.5);
DO(n,N1) Y[n]=.1*(n-N/2-.5);
//DO(n,N1) Y[n]=sy*sinh(.1*(n-N/2));

for(m=-3;m<4;m++) {M(m,-3)L(m,3)}
for(n=-3;n<4;n++) {M(-3,n)L(3,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
//for(m=0;m<M1;m+=10){ M(X[m],Y[0]);L(X[m],Y[N]);}
//for(n=0;n<N1;n+=5){ M(X[0],Y[n]);L(X[M],Y[n]);} fprintf(o,".003 W 0 1 0 RGB S\n");

z_type cu,cd;
DO(m,M1){ x=X[m];
DO(n,N1){ y=Y[n]; z=z_type(x,y); //c=F3(z-1.);//c=log(c);
        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=Zo*cu+Zc*cd;
        c=log(c);
        g[m*N1+n]=Re(c);
        f[m*N1+n]=Im(c); }}

fprintf(o,"1 setlinejoin 1 setlinecap\n");
DB p=1;
                        conto(o,f,w,v,X,Y,M,N, -3.,-3 ,3);fprintf(o,".04 W 1 .5 0 RGB S\n");
for(n=-28;n<-21;n+=2) conto(o,f,w,v,X,Y,M,N,.1*n,-p ,p);fprintf(o,".01 W 0 1 0 RGB S\n");
                        conto(o,f,w,v,X,Y,M,N, -2.,-1 ,1);fprintf(o,".04 W 1 .5 0 RGB S\n");
for(n=-18;n<-11;n+=2) conto(o,f,w,v,X,Y,M,N,.1*n,-p ,p);fprintf(o,".01 W 0 1 0 RGB S\n");
                        conto(o,f,w,v,X,Y,M,N, -1.,-1 ,1);fprintf(o,".04 W 1 .5 0 RGB S\n");
for(n= -8;n< 0;n+=2) conto(o,f,w,v,X,Y,M,N,.1*n,-p ,p);fprintf(o,".01 W 0 1 0 RGB S\n");
                        conto(o,f,w,v,X,Y,M,N, 0.,-1 ,1);fprintf(o,".04 W 0 .8 0 RGB S\n");
for(n= 2;n< 9;n+=2) conto(o,f,w,v,X,Y,M,N,.1*n,-p ,p);fprintf(o,".01 W 0 1 0 RGB S\n");
                        conto(o,f,w,v,X,Y,M,N, 1.,-1 ,1);fprintf(o,".04 W 0 .5 1 RGB S\n");
for(n= 12;n< 20;n+=2) conto(o,f,w,v,X,Y,M,N,.1*n,-p ,p);fprintf(o,".01 W 0 1 0 RGB S\n");
                        conto(o,f,w,v,X,Y,M,N, 2.,-1 ,1);fprintf(o,".04 W 0 .5 1 RGB S\n");
for(n= 22;n< 29;n+=2) conto(o,f,w,v,X,Y,M,N,.1*n,-p ,p);fprintf(o,".01 W 0 1 0 RGB S\n");
                        conto(o,f,w,v,X,Y,M,N, 3.,-3 ,3);fprintf(o,".04 W 0 .5 1 RGB S\n");

//p=.15;
// conto(o,g,w,v,X,Y,M,N,-.110551 ,-p ,p); fprintf(o,".01 W 1 0 1 RGB S\n");
// conto(o,g,w,v,X,Y,M,N, .175720,-p ,p); fprintf(o,".01 W 1 0 1 RGB S\n");
//p=20;
                        conto(o,g,w,v,X,Y,M,N, -4.,-1 ,1);fprintf(o,".04 W 1 0 0 RGB S\n");
for(n=-38;n<-31;n+=2) conto(o,g,w,v,X,Y,M,N,.1*n,-p ,p);fprintf(o,".01 W 1 0 0 RGB S\n");
                        conto(o,g,w,v,X,Y,M,N, -3.,-1 ,1);fprintf(o,".04 W 1 0 0 RGB S\n");
for(n=-28;n<-21;n+=2) conto(o,g,w,v,X,Y,M,N,.1*n,-p ,p);fprintf(o,".01 W 1 0 0 RGB S\n");
                        conto(o,g,w,v,X,Y,M,N, -2.,-1 ,1);fprintf(o,".04 W 1 0 0 RGB S\n");
for(n=-18;n<-11;n+=2) conto(o,g,w,v,X,Y,M,N,.1*n,-p ,p);fprintf(o,".01 W 1 0 0 RGB S\n");
                        conto(o,g,w,v,X,Y,M,N, -1.,-1 ,1);fprintf(o,".04 W 1 0 0 RGB S\n");
for(n= -8;n< 0;n+=2) conto(o,g,w,v,X,Y,M,N,.1*n,-p
,p);fprintf(o,".01 W 1 0 0 RGB S\n");
                        conto(o,g,w,v,X,Y,M,N, 0.,-1 ,1);fprintf(o,".05 W .5 0 .5 RGB S\n");
for(n= 2;n< 9;n+=2) conto(o,g,w,v,X,Y,M,N,.1*n,-p ,p);fprintf(o,".01 W 0 0 1 RGB S\n");
                        conto(o,g,w,v,X,Y,M,N, 1.,-1 ,1);fprintf(o,".04 W 0 0 1 RGB S\n");

fprintf(o,"0 setlinejoin 0 setlinecap\n");
M(-4, 3)L(-1, 3)M(1, 3)L(4, 3);
M(-4,-3)L(-1,-3)M(1,-3)L(4,-3); fprintf(o,".1 W 0 0 0 RGB S\n");
for(n=-38;n<-4;n+=5){M(.1*n-.25,0)L(.1*n+.05,0)}
fprintf(o,".08 W 1 0 1 RGB S\n");

fprintf(o,"showpage\n %c%cTrailer",'%','%'); fclose(o);
//system( "ggv fig02a.eps");
system("epstopdf analuxp02a.eps");
system( "open analuxp02a.pdf");
getchar();
system("killall Preview");
}

Latex generator of labels

% The C++ generators of the 3 pictures used are pretty similar, so the only one of them is copypasted above. However, all three output files are necessary to compile the Latex document below:

\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphics}
\usepackage{rotating}
\paperwidth 270pt
\paperheight 104pt
\topmargin -109pt
\oddsidemargin -72pt
\parindent 0pt
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}

\newcommand \analuka {
\put(7,89){\sx{.5}{$y$}}
\put(7,80.8){\sx{.5}{$3$}}
\put(7,70.8){\sx{.5}{$2$}}
\put(7,59.8){\sx{.5}{$1$}}
\put(7,49.8){\sx{.5}{$0$}}
\put(3,39.8){\sx{.5}{$-1$}}
\put(3,29.8){\sx{.5}{$-2$}}
\put(3,19.8){\sx{.5}{$-3$}}
\put(17.2,17){\sx{.5}{$-2$}}
\put(27.2,17){\sx{.5}{$-1$}}
\put(39.6,17){\sx{.5}{$0$}}
\put(49.7,17){\sx{.5}{$1$}}
\put(59.7,17){\sx{.5}{$2$}}
\put(69.7,17){\sx{.5}{$3$}}
\put(77.7,17){\sx{.5}{$x$}}
}
\sx{1.0}{\begin{picture}(90,104)
%%\put(0,0){\includegraphics{fig04}}
\put(0,0){\includegraphics{analuxp02a}}
\put(59,90){$A\!=\!3$}
\put(36,92){\sx{.46}{\rot{0.}$\rho\!=\!0$\ero}}
\put(36,10){\sx{.46}{\rot{0.}$\rho\!=\!0$\ero}}
\put(56,86){\sx{.46}{\rot{65}$\rho\!=\!0.2$\ero}}
\put(35,75){\sx{.46}{\rot{-6.}$\rho\!=\!-1$\ero}}
\put(14,63){\sx{.46}{\rot{76.}$\varphi\!=\!3$\ero}}
\put(32,59){\sx{.46}{\rot{84.}$\varphi\!=\!2$\ero}}
\put(39.6,58){\sx{.46}{\rot{66.}$\varphi\!=\!1$\ero}}
\put(35,36.6){\sx{.46}{\rot{21.}$\rho\!=\!-2$\ero}}
\put(35,26){\sx{.46}{\rot{8.}$\rho\!=\!-1$\ero}}
\put(66,64){\sx{.44}{\rot{60}$\rho\!=\!-1.6$\ero}}
\put(60,51){\sx{.44}{$\varphi\!=\!0$}}
\put(65.6,38.6){\sx{.44}{\rot{76}$\varphi\!=\!0$\ero}}
\analuka
\end{picture}}
\sx{1.0}{\begin{picture}(90,104)
\put(0,0){\includegraphics{analuxp02b}}
\put(59,90){$A\!=\!5$}
\put(36,94){\sx{.46}{\rot{-1.}$\rho\!=\!-1$\ero}}
\put(36.6,79.2){\sx{.46}{\rot{-12.}$\rho\!=\!-2$\ero}}
\put(26.4,62){\sx{.46}{\rot{90.}$\varphi\!=\!2$\ero}}
\put(35.2,59){\sx{.44}{\rot{60.}$\varphi\!=\!1$\ero}}
\put(34,36.6){\sx{.46}{\rot{38.}$\rho\!=\!-3$\ero}}
\put(35,22){\sx{.46}{\rot{8.}$\rho\!=\!-2$\ero}}
\put(68,51){\sx{.44}{$\varphi\!=\!0$}}
\put(76.8,26){\sx{.44}{\rot{69}$\varphi\!=\!0$\ero}}
%%\put(0,0){\includegraphics{fig04}}
\analuka
\end{picture}}
\sx{1.0}{\begin{picture}(90,104)
\put(0,0){\includegraphics{analuxp02c}}
\put(50,92){$A\!=\!10$}
\put(14,60){\sx{.46}{\rot{92.}$\varphi\!=\!2$\ero}}
\put(23.4,59){\sx{.44}{\rot{55.}$\varphi\!=\!1$\ero}}
\put(33,35.6){\sx{.46}{\rot{58.}$\rho\!=\!-4$\ero}}
\put(70.8,51){\sx{.44}{$\varphi\!=\!0$}}
\put(61,30){\sx{.44}{\rot{54}$\rho\!=\!-3.2$\ero}}
\put(71,23){\sx{.44}{\rot{40}$\rho\!=\!-3$\ero}}
\analuka
\end{picture}}
\end{document}

References

↑ https://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862
https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3
https://mizugadro.mydns.jp/BOOK/458.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Piblishing, 2020. Page 187, Fig.14.5.
↑ https://mizugadro.mydns.jp/BOOK/2020.pdf Дмитрий Кузнецов. Суперфункции. Lambert Academic Piblishing, 2014. Page 187, Fig.14.5.
↑ ^3.0 ^3.1 http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.

Keywords

«Exponential», «[[]]», «Superfunctions», «Tetration»,

File history

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

	Date/Time	Thumbnail	Dimensions	User	Comment
current	06:10, 1 December 2018		3,362 × 1,295 (987 KB)	Maintenance script (talk \| contribs)	Importing image file

You cannot overwrite this file.

File usage

There are no pages that use 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.

Horizontal resolution	900 dpi
Vertical resolution	900 dpi

[1] ttps://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862
https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3
https://mizugadro.mydns.jp/BOOK/458.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Piblishing, 2020. Page 187, Fig.14.5.

[2] ttps://mizugadro.mydns.jp/BOOK/2020.pdf Дмитрий Кузнецов. Суперфункции. Lambert Academic Piblishing, 2014. Page 187, Fig.14.5.

[analuxp-3] 3.0 ^3.1 http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.

[1]

[2]

[3]