File:Ausintay40t50.jpg

Complex map of the truncated Taulor series of function AuSin expanded at point $\pi/2$ with 40 terms taken into account.

C++ generator of curves
Files ado.cin, arcsin.cin, susin.cin, conto.cin should be loaded in order to compile the code below:

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"
 * 5) include "arcsin.cin"
 * 6) include"susin.cin"

z_type ausin0(z_type z){ z_type b,d,L; int n,N; DB c[32]= { 0.075238095238095238,0.011047619047619048, 0.0025161272589844018,0.00064214089261708309, 0.00015466661256774635,0.000027370601208859475, -2.7882266241028593e-7,-2.6398530640882673e-6, -8.4436657963874166e-7,2.1258409455641387e-7, 2.9625812021061085e-7,5.5084490596214015e-8, -8.505934497279864e-8,-5.7880498792647227e-8, 1.931195823640328e-8,4.1375071763898608e-8, 3.8895251916390044e-9,-2.9644790466835329e-8, -1.5302998576174588e-8,2.2013832459456463e-8, 2.5112957261097041e-8,-1.5610882396047076e-8, -3.8703209933396839e-8,6.3863715618218346e-9, 6.2295996364195277e-8,1.4512488433891138e-8, -1.0741668102638471e-7,-7.2006309159592681e-8, 1.9825395033112708e-7,2.4402848447224951e-7, -3.8457536962314424e-7,-7.917263057277091e-7}; N=21; L=log(z); b=z*z; //d=b*(c[N]*.5); d=b*c[N]*.5; for(n=N;n>=0;n--) { d+=c[n]; d*=b;} //d=b*(79./1050.+ b*(.29/2635.)); d+=1.2*L+3./b; //return d-    2.08962271972955094; //return d-    2.089622719729546; return d-      2.089622719729524; }

z_type ausintay40(z_type z){ int n,N; z_type s; DB b[44]={ 0.000000000000018573,2.291638074409576120,1.960438524396879600,1.078628512561472519, 0.596229979933950216,0.283339971398291091,0.141932611945484954,0.064237342712339371, 0.030266877055075078,0.013517212504274366,0.006114045543095120,0.002694686480147357, 0.001196608793087548,0.000520276081972018,0.000227565108909431,0.000098384760581466, 0.000042506775634220,0.000018246439059372,0.000007830495651859,0.000003341567773204, 0.000001424791217293,0.000000605357494037,0.000000256858981460,0.000000108708480622, 0.000000045942688496,0.000000019376065191,0.000000008162520256,0.000000003432848149, 0.000000001441940229,0.000000000604877447,0.000000000253477392,0.000000000106098816, 0.000000000044365678,0.000000000018533441,0.000000000007734554,0.000000000003225125, 0.000000000001345236,0.000000000000558311,0.000000000000233047,0.000000000000095023, 0.000000000000040981,0.000000000000018521,0.000000000000010568,0.000000000000005637}; N=40; //720 points were used for Cauchi; last 3 coefficients are poor; last 4 digits are wrong. z-=M_PI/2; z*=z; s=b[N]*z; for(n=N-1;n>0;n--) { s+=b[n]; s*=z; } return s; }

z_type ausin(z_type z){ int m,M; if(Im(z)<0) return conj(ausin(conj(z))); if(abs(z-M_PI/2)<1) return ausintay40(z); //M=50; DO(m,M) z=sin(z); M=50; DO(m,M) z=sin(z); return ausin0(z)-(0.+M); }

// #include "ausin.cin"

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; printf("1.-ausin(1)=%19.17f, 1.-ausin(z_type(1.,1.e-14))=%19.17f ;1.- susun(1.)=%19.17f, 1.-susun(z_type(1.,1.e-14))=%19.17f \n",     1.-Re(ausin(1.)), 1.-Re(ausin(z_type(1.,1.e-14))), 1.-Re(susin(1.)), 1.-Re(susin(z_type(1.,1.e-14))) ); //getchar;

int M=548,M1=M+1; int N=421,N1=N+1; DB X[M1],Y[N1]; DB *g, *f, *w; // w is working array. g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB))); f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB))); w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB))); char v[M1*N1]; // v is working array FILE *o;o=fopen("ausintay40.eps","w"); ado(o,332,402); fprintf(o,"1 201 translate\n 100 100 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); DB e; e=2./sinh(1.); DO(m,M1) { t=(m+.5)/400.; X[m]=e*sinh(1.*t);} e=2./sinh(1.); DO(n,N1) { t=(n-210.5)/210.; Y[n]=e*sinh(1.*t);} for(m=0;m<4;m+=1){M(m,-2) L(m,2) } for(n=-2;n<3;n+=1){M(0,n) L(M_PI,n)} fprintf(o,".002 W 0 0 0 RGB S\n"); M(M_PI/2,-2)L(M_PI/2,2) M(M_PI,-2)L(M_PI,2) fprintf(o,".002 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]; if(m/10*10==m) printf("x=%6.3f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.019) // c=arcsin(z); // c=sqrt(3./z); // c=susin(z); c=ausintay40(z); // d=susin(c); // d=arcsin(su0(z+1.)); // p=abs(z-d)/(abs(z)+abs(d)); p=-log(p)/log(10.); if(p>0 && p<17) g[m*N1+n]=p; p=Re(c); q=Im(c); if(p>-101 && p<101 && q>-101 && q<101){ g[m*N1+n]=p;f[m*N1+n]=q;} }} fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=10.;q=1.; /* p=9;q=.16; conto(o,g,w,v,X,Y,M,N,(15.6 ),-p,p); fprintf(o,".001 W .4 1 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(15. ),-p,p); fprintf(o,".005 W 1 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(14. ),-p,p); fprintf(o,".002 W .2 .2 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(13. ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(12. ),-p,p); fprintf(o,".006 W 0 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(11. ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(10. ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (9. ),-p,p); fprintf(o,".006 W 0 1 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (8. ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (7. ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (6. ),-p,p); fprintf(o,".006 W 0 .5 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (5. ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (4. ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (3. ),-p,p); fprintf(o,".006 W 1 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (2. ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (1. ),-p,p); fprintf(o,".004 W .5 0 0 RGB S\n"); for(m=-4;m<4;m++)for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".0016 W 0 .6 0 RGB S\n"); for(m=0;m<4;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".0016 W .9 0 0 RGB S\n"); for(m=0;m<4;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".0016 W 0 0 .9 RGB S\n"); for(m= 1;m<8;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".004 W .8 0 0 RGB S\n"); for(m= 1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".004 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".004 W .5 0 .5 RGB S\n"); for(m=-8;m<9;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".004 W 0 0 0 RGB S\n"); fprintf(o,"0 setlinejoin 0 setlinecap\n"); M(-10,0)L(0,0) fprintf(o,"1 1 1 RGB .01 W S\n"); DO(n,51) {x=-.2*n; M(x-.01,0) L(x-.09,0) } fprintf(o,"0 0 0 RGB .03 W S\n"); //#include "plofu.cin"

fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); free(f); free(g); free(w); system("epstopdf ausintay40.eps"); system( "open ausintay40.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphics} \usepackage{rotating} \paperwidth 3288pt \paperheight 4150pt \topmargin -100pt \oddsidemargin -72pt \textwidth 3200pt \textheight 4200pt \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \pagestyle{empty} \begin{document} \sx{10}{\begin{picture}(328,412) \put(10,9){\includegraphics{ausintay40}} \put(2,406){\sx{1.2}{$y$}} \put(2,306){\sx{1.2}{$1$}} \put(2,206){\sx{1.2}{$0$}} \put(-7,106){\sx{1.2}{$-1$}} %\put(-7, 06){\sx{1.2}{$-2$}} \put(9,-1){\sx{1.2}{$0$}} \put(109,-1){\sx{1.2}{$1$}} \put(157,-1){\sx{1.2}{$\pi/2$}} \put(209,-1){\sx{1.2}{$2$}} \put(309,-1){\sx{1.2}{$3$}} \put(319,-1){\sx{1.2}{$x$}} \put(184,208){$v\!=\!0$} \put(98,200){\rot{90}$u\!=\!2$\ero} \put(114,200){\rot{90}$u\!=\!1$\ero} \put(147,186){\rot{41}$u\!=\!0$\ero} \put(136,131){\rot{-40}$u\!=\!-1$\ero} \put(125,93){\rot{23}$v\!=\!0$\ero} \put(92,288){\rot{66}$u\!=\!-2$\ero} \put(131,282){\rot{31}$u\!=\!-1$\ero} %\put(159,244){\rot{90}$v\!=\!-0.2$\ero} %\put(172,250){\rot{90}$v\!=\!0$\ero} \put(148,260){$u\!=\!-0.5$} %\put(186,250){\rot{90}$v\!=\!0.2$\ero} \put(86,158){\rot{90}$v\!=\!3$\ero} \put(100,155){\rot{90}$v\!=\!2$\ero} \put(122,148){\rot{90}$v\!=\!1$\ero} \put(159,144){\rot{90}$v\!=\!0.2$\ero} \put(172,144){\rot{90}$v\!=\!0$\ero} \put(186,138){\rot{90}$v\!=\!-0.2$\ero} \end{picture}} \end{document}

Cateogry:Latex