File:Analuxp02t900.jpg

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.

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 "ado.cin" //#include"advacon.cin" //#include"f3c.cin" int main{ int m,n,j; DB x,y, A,Ay; z_type c,z;
 * 1) include
 * 2) define Re(z) z.real
 * 3) define Im(z) z.imag
 * 4) define z_type complex
 * 5) define DB double
 * 6) define DO(x,y) for(x=0;x<y;x++)
 * 1) include"conto.cin"
 * 1) define Zo z_type(.31813150520476413, 1.3372357014306895)
 * 2) define Zc z_type(.31813150520476413,-1.3372357014306895)
 * 3) define I z_type(0.,1.)

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}