File:AsincmapT.png

Complex map of function ArcSinc.

$u+\mathrm i v=\mathrm{asinc}(x+\mathrm i y)$

C++ implementation as asinc.cin
The text below is supposed to be stored as asinc.cin

z_type acos(z_type z){ if(Im(z)<0){if(Re(z)>=0){return I*log( z + sqrt(z*z-1.) );} else{return I*log( z - sqrt(z*z-1.) );}} if(Re(z)>=0){return -I*log( z + sqrt(z*z-1.) );} else {return -I*log( z - sqrt(z*z-1.) );} }

z_type asin(z_type z){ if(Im(z)<0){if(Re(z)>=0){return M_PI/2.-I*log( z + sqrt(z*z-1.) );} else {return M_PI/2.-I*log( z - sqrt(z*z-1.) );}} if(Re(z)>=0){return M_PI/2.+I*log( z + sqrt(z*z-1.) );} else {return M_PI/2.+I*log( z - sqrt(z*z-1.) );} }

z_type sinc(z_type z){ DB x=Re(z); DB y=Im(z); if(x*x+y*y>.01) return sin(z)/z; z*=z; return 1.+z*(-1./6.+z*(1./120.+z*(-1./5040+z*(1./362880.)))); }

z_type sincp(z_type z){ DB x=Re(z); DB y=Im(z); z_type c; if(x*x+y*y>.01) return (cos(z)-sinc(z))/z; c=z*z; return z*(-1./3.+c*(1./30.+c*(-1./840.+c*(1./45630+c*(-1./399168.))))); }

DB Namihei=4.493409457909062; DB Katsuo=-0.21723362821122164; z_type asincL(z_type z){int n; z_type s=Namihei-sqrt((-2./Katsuo)*(z-Katsuo)); DO(n,6)s+=(z-sinc(s))/sincp(s); return s;}

z_type asincU(z_type z){int n; z_type s=1.4*acos(z); DO(n,6)s+=(z-sinc(s))/sincp(s); return s;}

z_type asinc(z_type z){DB x=Re(z),y=fabs(Im(z)); if(y<1. && x<0. && x>-2.6 ) return asincL(z); return asincU(z); }

C++ generator of curves
// File asinc.cin above, and also files ado.cin and conto.cin

// sould be loaded in the working directory in order to compile the C++ code below.

using namespace std; typedef complex z_type;
 * 1) include 
 * 2) include 
 * 3) define DB double
 * 4) 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"
 * 5) include "asinc.cin"

main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=801,M1=M+1; int N=401,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("asincmap.eps","w");ado(o,82,82); fprintf(o,"41 41 translate\n 10 10 scale\n"); DO(m,400) X[m]=-4.+.01*m; X[400]=-.002; X[401]= .002; for(m=402;m<M1;m++) X[m]=-4.+.01*(m-1.); DO(n,200)Y[n]=-4.+.02*n; Y[200]=-.002; Y[201]= .002; for(n=202;n-99. && p<99.      &&     q>-99. && q<99       ) {g[m*N1+n]=p; f[m*N1+n]=q; }                     }} // #include "plodi.cin" /* fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=22; conto(o,g,w,v,X,Y,M,N, (15.4 ),-p,p); fprintf(o,".02 W 0 .6 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (15. ),-p,p); fprintf(o,".02 W 1 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (14. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (13. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (12. ),-p,p); fprintf(o,".02 W 0 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (11. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (10. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (9. ),-p,p); fprintf(o,".02 W 0 1 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (8. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (7. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (6. ),-p,p); fprintf(o,".02 W 1 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (5. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (4. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (3. ),-p,p); fprintf(o,".03 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (2. ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (1. ),-p,p); fprintf(o,".02 W .6 0 0 RGB S\n"); */ fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.5;q=.5; for(m=-11;m<11;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".007 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,".007 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,".007 W 0 0 .9 RGB S\n"); for(m=1;m<11;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".02 W .9 0 0 RGB S\n"); for(m=1;m<11;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".02 W .6 0 .6 RGB S\n"); for(m=-9;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 W 0 0 0 RGB S\n"); m=0;         conto(o,g,w,v,X,Y,M,N, (0.+m),-2*p,2*p); fprintf(o,".02 W 0 0 0 RGB S\n"); for(m=1;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 W 0 0 0 RGB S\n"); //#include "plofu.cin" M(-4.01,0)L(Katsuo,0)M(1,0)L(4.01,0) fprintf(o,"0 setlinecap .021 W 1 1 1 RGB S\n"); DO(n,39){M(Katsuo-.1*n,0) L(Katsuo-.1*(n+.5),0)} DO(n,31){M(1+.1*n,0) L(1+.1*(n+.5),0)} fprintf(o,"0 setlinecap .024 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf asincmap.eps"); system(   "open asincmap.pdf"); getchar; system("killall Preview");//for mac }

Latex generator of labels
% File asincmap.pdf should be generated with the code above in order to compile the Latex document below.

% \documentclass[12pt]{article} % \paperheight 838px % \paperwidth 844px % \textwidth 1294px % \textheight 1200px % \topmargin -80px % \oddsidemargin -80px % \usepackage{graphics} % \usepackage{rotating} % \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \newcommand \rmi {\mathrm{i}} % \begin{document} % \newcommand \zoomax { % \put(16,820){\sx{4.4}{$y$}} % \put(16,630){\sx{4}{$2$}} % \put(16,430){\sx{4}{$0$}} % \put(-4, 230){\sx{4}{$-\!2$}} % \put(220, 5){\sx{4}{$-\!2$}} % \put(443, 5){\sx{4}{$0$}} % \put(643, 5){\sx{4}{$2$}} % \put(831,6){\sx{4}{$x$}} % } % \parindent 0pt % %\sx{8}{\begin{picture}(86,86) \put(0,0){\ing{b271t0}} % \begin{picture}(816,816) % \put(40,30){\sx{10}{\ing{asincmap}}} % \zoomax % \put(100,548){\sx{4}{\rot{-11}$u\!=\!3.6$\ero}} % \put(90,484){\sx{4}{\rot{-1}$u\!=\!3.8$\ero}} % \put(90,375){\sx{4}{\rot{1}$u\!=\!3.8$\ero}} % \put(209,150){\sx{4}{\rot{40}$u\!=\!3$\ero}} % \put(501, 40){\sx{4}{\rot{86}$u\!=\!2$\ero}} % % \put(654,634){\sx{4}{\rot{56}$u\!=\!1.2$\ero}} % \put(686,610){\sx{4}{\rot{47}$u\!=\!1$\ero}} % \put(698,574){\sx{4}{\rot{36}$u\!=\!0.8$\ero}} % \put(720,540){\sx{4}{\rot{26}$u\!=\!0.6$\ero}} % \put(722,500){\sx{4}{\rot{20}$u\!=\!0.4$\ero}} % \put(720,463){\sx{4}{\rot{10}$u\!=\!0.2$\ero}} % % \put(70,432){\sx{4}{\bf cut}} % \put(690,432){\sx{4}{\bf cut}} % \put(436,703){\sx{4}{$v\!=\!-3$}} % \put(432,550){\sx{4}{$v\!=\!-2$}} % \put(441,430){\sx{4}{$v\!=\!0$}} % \put(450,385){\sx{4}{$v\!=\!1$}} % \put(452,307){\sx{4}{$v\!=\!2$}} % \put(452,153){\sx{4}{$v\!=\!3$}} % \end{picture} % \end{document} % % %