File:ExpMapT.png

Complex map of the exponential function.

$u\!+\!\mathrm i v=\exp(x\!+\!\mathrm i y)$.

C++ generator of curves

Files ado.cin and conto.cin should be loaded in order to compile the code below

#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
using namespace std;
#include<complex>
typedef complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
//#include "fsexp.cin"
//#include "fslog.cin"
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
int M=401,M1=M+1;
int N=401,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("ExpMap.eps","w");  ado(o,802,802);
fprintf(o,"401 401 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-4.+.02*(m-.5);
DO(n,N1) Y[n]=-4.+.02*(n-.5); 
// for(n=0;n<N1;n++) Y[n]=sinh((2./200.)*(n-200.5));
for(m=-4;m<5;m++) {M(m,-4)L(m,4)}
for(n=-4;n<5;n++) {M(  -4,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
//fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n");
DO(m,M1)DO(n,N1){      g[m*N1+n]=999;
                       f[m*N1+n]=999;}
DB b=sqrt(2);
DO(m,M1){x=X[m]; printf("x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.019)
      { 
       c=exp(z);
//     c=FSEXP(z);
//     c=FSLOG(z);
        p=Re(c); q=Im(c);
       // if(p>-12 && p<12 && fabs(q)>1.e-12) 
       g[m*N1+n]=p;
       // if(q>-12 && q<12 && fabs(q)>1.e-12) 
       f[m*N1+n]=q;
      }
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");
p=2.;q=.3;
for(m=-8;m<8;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<8;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<8;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<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".02 W .8 0 0 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".02 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".02 W .5 0 .5 RGB S\n");
for(m=-16;m<17;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(-2,0)L(-4,0)fprintf(o,"0 setlinecap .03 W 1 1 1 RGB S\n");
for(n=0;n<11;n++){ M(-2-.2*(n+.2),0) L(-2-.2*(n+.45),0) } fprintf(o,".04 W 1 .5 0 RGB S\n");
for(n=0;n<21;n++){ M(-2-.2*(n+.7),0) L(-2-.2*(n+.95),0) } fprintf(o,".04 W 0 .5 1 RGB S\n");
*/
//#include "plofu.cin"
fprintf(o,"0 setlinejoin 0 setlinecap\n");
fprintf(o,"showpage\n");
fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o);
free(f);
free(g);
free(w);
      system("epstopdf ExpMap.eps"); 
      system(    "open ExpMap.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex generator of labels

% %<br> \documentclass[12pt]{article} % <br> \paperheight 828px % <br> \paperwidth 836px % <br> \textwidth 1294px % <br> \textheight 1100px % <br> \topmargin -104px % <br> \oddsidemargin -78px % <br> \usepackage{graphics} % <br> \usepackage{rotating} % <br> \newcommand \sx {\scalebox} % <br> \newcommand \rot {\begin{rotate}} % <br> \newcommand \ero {\end{rotate}} % <br> \newcommand \ing {\includegraphics} % <br> \newcommand \rmi {\mathrm{i}} % <br> \begin{document} % <br> \newcommand \zoomax { % <br> \put(18, 806){\sx{3.3}{$y$}} % <br> \put(18, 713){\sx{3}{$3$}} % <br> \put(18, 613){\sx{3}{$2$}} % <br> \put(18, 513){\sx{3}{$1$}} % <br> \put(18, 413){\sx{3}{$0$}} % <br> \put(-6, 313){\sx{3}{$-1$}} % <br> \put(-6, 213){\sx{3}{$-2$}} % <br> \put(-6, 113){\sx{3}{$-3$}} % <br> \put(-6, 013){\sx{3}{$-4$}} % <br> \put(014, -5){\sx{3}{$-4$}} % <br> \put(114, -5){\sx{3}{$-3$}} % <br> \put(214, -5){\sx{3}{$-2$}} % <br> \put(314, -5){\sx{3}{$-1$}} % <br> \put(435, -5){\sx{3}{$0$}} % <br> \put(535, -5){\sx{3}{$1$}} % <br> \put(635, -5){\sx{3}{$2$}} % <br> \put(735, -5){\sx{3}{$3$}} % <br> \put(827,-4){\sx{3}{$x$}} % <br> } % <br> \parindent 0pt % <br> \sx{1}{\begin{picture}(852,820) % <br> %\put(40,20){\ing{b271tMap3}} % <br> \put(40,20){\ing{ExpMap}} % <br> \zoomax % <br> \put(100,726){\sx{4}{$v\!=\!0$}} % <br> \put(100,568){\sx{4}{$u\!=\!0$}} % <br> \put(100,411){\sx{4}{$v\!=\!0$}} % <br> \put(100,252){\sx{4}{$u\!=\!0$}} % <br> \put(100,094){\sx{4}{$v\!=\!0$}} % <br> % <br> \put(455,677){\sx{4}{\rot{90}$u\!=\!-1$\ero}} % <br> % <br> \put(455,532){\sx{4}{\rot{90}$v\!=\!1$\ero}} % <br> \put(525,532){\sx{4}{\rot{90}$v\!=\!2$\ero}} % <br> % <br> \put(296,360){\sx{4}{\rot{90}$u\!=\!0.2$\ero}} % <br> \put(455,374){\sx{4}{\rot{90}$u\!=\!1$\ero}} % <br> \put(525,374){\sx{4}{\rot{90}$u\!=\!2$\ero}} % <br> % <br> \put(455,208){\sx{4}{\rot{90}$v\!=\!-\!1$\ero}} % <br> \put(455,050){\sx{4}{\rot{90}$u\!=\!-\!1$\ero}} % <br> \end{picture}} % <br> \end{document} % <br>

References