File:Vladi10.jpg

Complex map of approximation slo of the natural arctetration ate and agreements at the substitution of this approximation into the transfer equation.

Usage: this is figure 15.2 of the book Суперфункции (2014, In Russian) ; the English version is in preparation in 2015.

First time published in the Vladikavkaz Matehmatical Journal .

C++ generator of the compelx map
ado.cin, conto.cin, fslog.cin should be loaded in order to compile the code below.

typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; z_type Zo=z_type(.31813150520476413, 1.3372357014306895); z_type Zc=z_type(.31813150520476413,-1.3372357014306895);
 * 1) include "fslog.cin"
 * 2) include "conto.cin"

int M=200,M1=M+1; int N=402,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 //DB X[201],Y[403], g[40803],f[40803], w[40803]; // w is working array. DB X[201],Y[403], g[81003],f[81003], w[81003]; // w is working array. char v[81003]; // v is working array //FILE *o;o=fopen("figslo.eps","w");ado(o,0,0,62,62); FILE *o;o=fopen("vladi10a.eps","w");ado(o,62,62); fprintf(o,"21 31 translate\n 10 10 scale\n");

z_type L=z_type(.31813150520476413, 1.3372357014306895); p=Re(L); q=Im(L); DB R=abs(L); DB A=arg(L); fprintf(o,"0 0 %9.6f %9.6f %9.6f arc C .5 1 0 RGB F\n",R,-180/M_PI*A,180/M_PI*A);

// DB sx=3./sinh(.01*N); // DO(m,M1) X[m]=sx*sinh(.02*(m-M/2-.5)); DO(m,M1) X[m]=-1.+.02*(m-.5); //DO(n,N1)Y[n]=-3.3+.014*(n-.5); DO(n,N1)Y[n]=-2.+.01*(n-.5); //DB sy=6./sinh(.01*N); //DO(n,N1) Y[n]=sy*sinh(.02*(n-N/2-.5));

for(m=-2;m<4;m++) {    if(m==0){M(m,-2.1)L(m,2.1)} else   {M(m,-2)L(m,2)}                 } for(n=-2;n<3;n++) {M( -2,n)L(3,n)} fprintf(o,".006 W 0 0 0 RGB S\n");

DO(m,M1)DO(n,N1){      g[m*N1+n]=9999; f[m*N1+n]=9999; } //z_type F[M1*N1]; DO(m,M1){x=X[m]; printf("50 run at x=%6.3f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); c=slo(z); p=Re(c); q=Im(c); if(p>-9 && p<9 && q>-9 && q<9 ) {g[m*N1+n]=p;f[m*N1+n]=q;} }} p=.5;q=.4; //p=2;q=1; fprintf(o,"1 setlinecap 1 setlinejoin\n");

for(m=-19;m<19;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".006 W 0 .6 0 RGB S\n"); for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q); fprintf(o,".006 W .9 0 0 RGB S\n"); for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".006 W 0 0 .9 RGB S\n");

for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".024 W .8 0 0 RGB S\n"); for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".024 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".024 W .5 0 .5 RGB S\n"); for(m=-31;m<32;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".024 W 0 0 0 RGB S\n");

//#include"plofu.cin"

fprintf(o,"0 setlinecap 0 setlinejoin\n");

M(Re(Zo),Im(Zo)) L(-2.02,Im(Zo)) M(Re(Zo),-Im(Zo)) L(-2.02,-Im(Zo)) fprintf(o,".02 W 1 1 1 RGB 0 setlinecap S\n"); DO(m,8){ M(Re(Zo)-.31*(m ), Im(Zo)) L(Re(Zo)-.31*(m+.5), Im(Zo)) } DO(m,8){ M(Re(Zo)-.31*(m ),-Im(Zo)) L(Re(Zo)-.31*(m+.5),-Im(Zo)) } fprintf(o,".05 W 0 0 0 RGB S\n"); // fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n"); //setdash is not supported fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf vladi10a.eps"); system(   "open vladi10a.pdf"); getchar; system("killall Preview"); }

C++ generator of nap of agreement $D_{\mathrm A}$
ado.cin, conto.cin, fslog.cin should be loaded in order to compile the code below.

typedef std::complex z_type; //#include  //#define z_type complex
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; z_type Zo=z_type(.31813150520476413, 1.3372357014306895); z_type Zc=z_type(.31813150520476413,-1.3372357014306895);
 * 1) include "fslog.cin"
 * 2) include "conto.cin"

int M=200,M1=M+1; int N=201,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("figsloE.eps","w");ado(o,0,0,52,52); FILE *o;o=fopen("vladi10b.eps","w");ado(o,52,52); fprintf(o,"21 31 translate\n 10 10 scale\n");

z_type L=z_type(.31813150520476413, 1.3372357014306895); p=Re(L); q=Im(L); DB R=abs(L); DB A=arg(L); fprintf(o,"0 0 %9.6f %9.6f %9.6f arc C .5 1 0 RGB F\n",R,-180/M_PI*A,180/M_PI*A);

DO(m,M1) X[m]=-1.2+.024*(m-.5); //DO(n,N1) Y[n]=-1.2 +.02*n*(1.+.000008*(n-8.)*(n-8)); DO(n,N1) Y[n]=-2.2 +.024*n;

//DB sy=6./sinh(.005*N); //DO(n,N1) Y[n]=sy*sinh(.01*(n-N/2-.5+10));

for(m=-2;m<4;m++) {    if(m==0){M(m,-2.1)L(m,2.08)} else   {M(m,-2)L(m,2)}                 } for(n=-2;n<3;n++) {M( -2,n)L(3,n)} fprintf(o,".006 W 0 0 0 RGB S\n");

DO(m,M1)DO(n,N1){      g[m*N1+n]=9999; f[m*N1+n]=9999; } for(m=0;m-99 && p<99) g[m*N1+n]=p; }} //M(-10,0)L(-2,0)fprintf(o,".04 W 0 0 0 RGB [.1 .1] 0 setdash S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf vladi10b.eps"); system(   "open vladi10b.pdf"); getchar; system("killall Preview"); }
 * 1) include"plodi.cin"

C++ generator of nap of agreement $D_{\mathrm B}$
ado.cin, conto.cin, fslog.cin should be loaded in order to compile the code below.

typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; z_type Zo=z_type(.31813150520476413, 1.3372357014306895); z_type Zc=z_type(.31813150520476413,-1.3372357014306895);
 * 1) include "fslog.cin"
 * 2) include "conto.cin"

int M=200,M1=M+1; int N=201,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("vladi10c.eps","w");ado(o,52,52); fprintf(o,"21 31 translate\n 10 10 scale\n");

z_type L=z_type(.31813150520476413, 1.3372357014306895); p=Re(L); q=Im(L); DB R=abs(L); DB A=arg(L); fprintf(o,"0 0 %9.6f %9.6f %9.6f arc C .5 1 0 RGB F\n",R,-180/M_PI*A,180/M_PI*A);

DO(m,M1) X[m]=-1.2+.024*(m-.5); //DO(n,N1) Y[n]=-1.2 +.02*n*(1.+.000008*(n-8.)*(n-8)); DO(n,N1) Y[n]=-2.2 +.024*n;

//DB sy=6./sinh(.005*N); //DO(n,N1) Y[n]=sy*sinh(.01*(n-N/2-.5+10));

for(m=-2;m<4;m++) {    if(m==0){M(m,-2.1)L(m,2.08)} else   {M(m,-2)L(m,2)}                 } for(n=-2;n<3;n++) {M( -2,n)L(3,n)} fprintf(o,".006 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("50 run at x=%6.3f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); c=slo(log(z))+1.; d=slo(z); p=abs(c-d); p=-log(p)/log(10.); // p=Re(log(c))/log(10.); if(p>-99 && p<99) g[m*N1+n]=p; }} fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf vladi10c.eps"); system(   "open vladi10c.pdf");//linux //getchar; system("killall Preview");//macintosh }
 * 1) include"plodi.cin"

Latex combiner
\documentclass[12pt]{article} \usepackage{graphicx} \usepackage{rotating} \usepackage{geometry} \paperwidth 418px %\paperheight 134px \paperheight 105px \topmargin -106pt \oddsidemargin -94pt \pagestyle{empty} \begin{document} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox}

\newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \fsloax { \put(-2.3,49){\sx{.45}{$y$}} \put(-2.3,39.6){\sx{.45}{$1$}} \put(-2.3,29.6){\sx{.45}{$0$}} \put(-6,19.6){\sx{.45}{$-1$}} \put(-6, 9.6){\sx{.45}{$-2$}} \put( 7 ,6.6){\sx{.45}{$-\!1$}} \put(19.8, 6.6){\sx{.45}{$0$}} \put(30, 6.6){\sx{.45}{$1$}} \put(40, 6.6){\sx{.45}{$2$}} \put(48.6, 6.6){\sx{.45}{$x$}} } \hskip 17pt %\sx{2.75}{\begin{picture}(60,55) \sx{2.34}{\begin{picture}(60,50) \put(0,0){\includegraphics{vladi10a}} \fsloax \put(32.2,25.8){\rot{90}\sx{.4}{$u\!=\!0$}\ero} \put(37,29.8){\sx{.4}{$v\!=\!0$}} \end{picture}} \sx{2.34}{\begin{picture}(60,50) %\put(0,0){\includegraphics{figsloE}} \put(0,0){\includegraphics{vladi10b}} \fsloax \put(32,44){\sx{.5}{$D_{\rm A}\!<\!1$}} \put(21,33){\sx{.55}{$15$}} \end{picture}} \sx{2.34}{\begin{picture}(50,50) %\put(0,0){\includegraphics{figsloL}} \put(0,0){\includegraphics{vladi10c}} \fsloax \put(3,33){\sx{.5}{$D_{\rm B}\!<\!1$}} \put(32,33){\sx{.55}{$15$}} \end{picture}}

\end{document}