File:TaniaNegMapT.png

Complex map of the truncation of the asymptotic expansion of the Tania function for the domain between the cut lines.

For $|\Im(z)| < \pi$, $\Re(z)\rightarrow - \infty$,


 * $ \mathrm{Tania}(z) ~ \sim ~

\varepsilon -\varepsilon^2+\frac{3}{2}\varepsilon^3 -\frac{8}{3}\varepsilon^4+\frac{125}{24}\varepsilon^6+ O(\varepsilon^6) $ where $\varepsilon=\exp(1+z)$

The truncation of the series gives the approximation shown in the figure.

The map shows
 * $ f=\varepsilon

-\varepsilon^2+\frac{3}{2}\varepsilon^3 -\frac{7}{2}\varepsilon^4 $

in the plane $x=\Re(z)$, $y=\Im(z)$; the lines $u=\Re(f)=\mathrm{const}$ and the lines $v=\Im(f)=\mathrm{const}$ are drawn.

The shading indicate the region with precision worse than 3 decimal digits

Common header
Files conto.cin and ado.cin should be in the working directory.

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

z_type ArcTania(z_type z) {return z + log(z) - 1. ;} z_type ArcTaniap(z_type z) {return 1. + 1./z ;}

z_type TaniaTay(z_type z) { int n; z_type s; s=1.+z*(.5+z*(1./16.+z*(-1./192.+z*(-1./3072.+z*(1.3/6144.+z*(-4.7/147456. //+z*(7.3/4128768.) //some reserve term )))))); DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }

z_type TaniaNega(z_type z){int n;z_type s=exp(z-exp(z)+1.); DO(n,4) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }

z_type TaniaNeg(z_type z){int n; z_type e=exp(1.+z); return e*(1.+e*(-1.+e*(1.5+e*(-8./3.+e*(125./24. ) )))); } z_type TaniaBig(z_type z){ int n; z_type t=1.+z; z_type L=log(t); z_type x=L/t; z_type m=1./L; z_type s = t-L + x*(1. + x*( .5-m + x*( 1./3. + m*(-1.5+m) +x*( .25 +m*(-11./6.+m*(3.-m)) //     +x*(.2 +m*(-25./12 +m*(35./6. +m*(-5. +m)))) //reserve term for the testing )))); //DO(n,2) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }

z_type TaniaBig0(z_type z){int n;z_type L=log(z), s=z-L+1.; s-=(1.-L)/z; return s ; DO(n,4) s+=(z-ArcTania(s))/ArcTaniap(s); }

z_type TaniaS(z_type z){int n; z_type s,t=z+z_type(2.,-M_PI);t*=2./9.; t=I*sqrt(t); s=-1.+t*(3.+t*(-3.+t*(.75+t*(.3+t*(.9/16.+t*(-.3/7.+t*(-12.51/224. //+t*(-.9/28.) ))))))); DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }

z_type Tania(z_type z){ z_type t; if( fabs(Im(z))< M_PI && Re(z)<-2.51) return TaniaNega(z); if( abs(z)>7. || Re(z)>3.8 ) return TaniaBig0(z); if( Im(z) > .7 ) return TaniaS(z); if( Im(z) < -.7) return conj(TaniaS(conj(z))); return TaniaTay(z); }

C++ generator of the shading
main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=160,M1=M+1; int N=160,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("tanianegmapD2.eps","w");ado(o,162,162); fprintf(o,"81 81 translate\n 10 10 scale\n"); DO(m,M1)X[m]=-8.+.1*(m); DO(n,N1)Y[n]=-8.+.1*(n); for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}} for(n=-8;n<9;n++){    M(  -8,n)L(8,n)} fprintf(o,".008 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("%5.2f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); c=TaniaNeg(z); d=Tania(z); // c=ArcTania(c); p=-log( abs(c-d) / (abs(c)+abs(d)) )/log(10.) ; //p=Re(c);q=Im(c); if(p>-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;} }} fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=16;q=.5; conto(o,g,w,v,X,Y,M,N, (3),-p,p); L(-8.,8) L(8,8) L(8,-8) L(-8,-8) fprintf(o,"1 .9 .9 RGB C F\n"); y= M_PI; for(m=0;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} y=-M_PI; for(m=0;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} fprintf(o,".07 W 1 .5 0 RGB S\n"); y= M_PI; for(m=2;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} y=-M_PI; for(m=2;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} fprintf(o,".07 W 0 .5 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf tanianegmapD2.eps"); system(   "open tanianegmapD2.pdf"); getchar; system("killall Preview"); }

C++ generator of the curves
main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=160,M1=M+1; int N=161,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("tanianegmap1.eps","w");ado(o,162,162); fprintf(o,"81 81 translate\n 10 10 scale\n"); DO(m,M1) X[m]=-8.+.1*(m); DO(n,80)Y[n]=-8.+.1*n; Y[80]=-.03; Y[81]= .03; for(n=82;n-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;} }} fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=.6;q=.5; for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".01 W 0 .6 0 RGB S\n"); for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".01 W .9 0 0 RGB S\n"); for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".01 W 0 0 .9 RGB S\n"); for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".05 W .9 0 0 RGB S\n"); for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".05 W .6 0 .6 RGB S\n"); for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 0 RGB S\n"); y= M_PI; for(m=0;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} y=-M_PI; for(m=0;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} fprintf(o,".07 W 1 .5 0 RGB S\n"); y= M_PI; for(m=2;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} y=-M_PI; for(m=2;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} fprintf(o,".07 W 0 .5 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf tanianegmap1.eps"); system(   "open tanianegmap1.pdf"); getchar; system("killall Preview"); }

Latex generator of the lables
% Gerenator of TaniaNegMap.png % Copyleft 2011 by Dmitrii Kouznetsov \documentclass[12pt]{article} % \usepackage{geometry} % \usepackage{graphicx} % \usepackage{rotating} % \paperwidth 854pt % \paperheight 844pt % \topmargin -96pt % \oddsidemargin -98pt % \textwidth 1100pt % \textheight 1100pt % \pagestyle {empty} % \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \begin{document} % \sx{5}{ \begin{picture}(164,165) % % \put(6,5){\ing{taniacontour}} % \put(6,5){\ing{taniaNegmapD2}} % \put(6,5){\ing{taniaNegmap1}} % \put(2,162){\sx{.7}{$y$}} % \put(2,144){\sx{.6}{$6$}} % \put(2,124){\sx{.6}{$4$}} % \put(2,104){\sx{.6}{$2$}} % %\put(3,116){\sx{.6}{$\pi$ ~ \bf cut}} % \put(23,100){\sx{.8}{$u\!=\!0$}} % %\put(2, 84){\sx{.6}{$0$}} % \put(20, 84){\sx{.8}{$v\!=\!0$}} % % \put(20, 84){\sx{.8}{\bf cut}} % \put(23,68){\sx{.8}{$u\!=\!0$}} % \put(-3,64){\sx{.6}{$-2$}} % %\put(-3,53){\sx{.6}{$-\pi$ ~ \bf cut}} % \put(-3,44){\sx{.6}{$-4$}} % \put(-3,24){\sx{.6}{$-6$}} % \put( 22,0){\sx{.6}{$-6$}} % \put( 42,0){\sx{.6}{$-4$}} % \put( 62,0){\sx{.6}{$-2$}} % \put( 86,0){\sx{.6}{$0$}} % \put(106,0){\sx{.6}{$2$}} % \put(126,0){\sx{.6}{$4$}} % \put(146,0){\sx{.6}{$6$}} % \put(164,0){\sx{.7}{$x$}} % % \put(139,154){\rot{-6}\sx{.8}{$v\!=\!6$}\ero}% % \put(138,143){\rot{-6}\sx{.8}{$v\!=\!5$}\ero}% % \put(137,132){\rot{-6}\sx{.8}{$v\!=\!4$}\ero}% % \put(136,121){\rot{-6}\sx{.8}{$v\!=\!3$}\ero}% % \put(135,109){\rot{-5}\sx{.8}{$v\!=\!2$}\ero}% % \put( 89, 83){\rot{86}\sx{.8}{$u\!=\!1$}\ero}% % \put(106, 77){\rot{87}\sx{.8}{$u\!=\!2$}\ero}% % \put(121, 77){\rot{88}\sx{.8}{$u\!=\!3$}\ero}% % \put(134, 97){\rot{-4}\sx{.8}{$v\!=\!1$}\ero}% % \put(134, 84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}% % \put(134, 72){\rot{3}\sx{.72}{$v\!=\!-\!1$}\ero}% % \put(133, 60){\rot{3}\sx{.72}{$v\!=\!-\!2$}\ero}% % \put(134, 48){\rot{3}\sx{.72}{$v\!=\!-\!3$}\ero}% % \put(135, 36){\rot{3}\sx{.72}{$v\!=\!-\!4$}\ero}% % \put(136, 25){\rot{3}\sx{.72}{$v\!=\!-\!5$}\ero}% % \put(137, 14){\rot{3}\sx{.72}{$v\!=\!-\!6$}\ero}% \end{picture} % } % \end{document}