File:Analuxp02t900.jpg

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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 2 by [1].

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

  1. ↑ http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html D.Kouznetsov. (2009). Solutions of F(z+1)=exp(F(z)) in the complex plane.. Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7.

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