File:Analuxp01t400.jpg

Comparison of various fits of the natural titration with the [[complex map]s

Lines of constant logamplitude $u$ and phase $v$ are shown in the complex plane:

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

for various approximations described below.

Description of curves
Note that in this figure, $u$ and $v$ are logamplitude and phase of the plotted functions; not the real and imaginary parts, as usually. This corresponds to the displacement of the map for unity to the right, along the real axis. In this case, it is easier to guess the asymptotic behaviour of the function (last picture, d) from its primitive fit (picture b).

a: Linear approximation by Gusmad
This is approximation, function $f$ linear in the ramge  $-1 < \Re(z) \le  0$. This approximation had been suggested in 2006 by M.H.Hooshmand .

$f(z)=\mathrm{uxp}(z)=\!\left\{\!\!\!\! \begin{array}{ccccc cc} \ln\!\big({\rm uxp}(z\!+\!1)\big)~&~{\rm at}~&~   \!&\! \!&\! \Re(z) \!&\! \le \!&\! -1 \\                    z  +  1      ~&{~\rm at}~&~ -1 \!&\! < \!&\! \Re(z) \!&\! \le \!&\! 0 \\ \exp\!\big({\rm uxp}(z\!-\!1)\big)~&{~\rm at}~&~ 0 \!&\! < \!&\! \Re(z) \!& \end{array} \right.$

b: Approximation for moderate values of imaginary part of the argument
$ f(z)=\mathrm{Fit}_{3}(z) = \left\{\!\! \begin{array}{ccccc cc} \ln\!\big({\rm Fit}_{3}(z\!+\!1)\big)~&~{~\rm at~}~&~  \!&\! \!&\! \Re(z) \!&\! \le \!&\! -1 \\               \mathrm{fit}_{3}(z)           ~&~{~\rm at~}~& -1 \!&\! < \!&\! \Re(z) \!&\! \le \!&\! 0 \\ \exp\!\big({\rm Fit}_{3}(z\!-\!1)\big)~&~{~\rm at~}~&~ 0 \!&\! < \!&\! \Re(z) \!& \end{array} \right. $

where

$\displaystyle \mathrm{fit}_3(z) \!=\! 0.6\!~\mathrm{fit}_{2}(z)+0.4\!~\ln\big( \mathrm{fit}_{2}(z+1)\big) $

$\displaystyle \mathrm{fit}_2(z) \!=\! \ln(2\!+\!z) + (1\!+\!z)\left(       1 +         \frac{z}{2}\exp\!\Big((z\!-\!1)s_2(z)\Big)        			\Big(\!\mathrm e\! -2\! +\! \ln\frac{4}{3}	\Big)		- \ln 2        \right)$

$ s_2(z) = \exp\!\Big(\exp(z-2.51)\Big)-0.6+0.08(z\!+\!1) $

c. Approximation for large values of imaginary part of the argument
In the upper half plane, say, $y>1/2$, the $\mathrm{fit}_6$ is shown,

$f(z)=\mathrm{fit}_6(z) = \left\{ \begin{array} ~ L+\exp(kz+r) ~, ~ \Re(z)<-8\\ \exp\Big(\mathrm{fit}_6(z\!-\!1)\Big)~,~ \Re(z)\ge -8 \end{array} \right.$

for $\Re(z)>0.5$ and

$f(z)=\mathrm{fit}_6(z^*)^*$

in the lower half plane, say, $\Im(z)<-1/2$.

The strip of intermediate values $|y|<1/2$ in the picture c is left empty.

In formula above, $L\approx 0.31813150520476413 + 1.3372357014306895 \,\mathrm i$ is fixed point of logarithm, $L=\ln(L)$. For Natural tetration, the increment $k=L$. Parameter $r$ provides the match of the two asymptotics. It is fundamental mathematical constant; $r \approx 1.075820830781 - 0.9466419207254 \, \mathrm i$. This precision seems to be sufficient for the applications; however, the improvement of the precision may be subject of the additional research.

d: Precise approximation.
The precise approximation of the natural tetration, with 14 decimal digits, is described at  . is this text seen? is this text seen?

The figure shown is almost the same, as in figure 1 in the first description of the real-holomorphic tetration to base e.

However, here, the labels are added to mark the curves.

Similar figure for real and imaginary parts of the functions is also available as http://mizugadro.mydns.jp/t/index.php?title=File:Analuxp01u400.jpg Due to the transfer equation

$\exp(f(z))=f(z\!+\!1)$

at moderate values of phase (smaller than $\pi$), the curves of the phase look as those for the imaginary part, displaced for -1 along the real axis; the same happen with curves for the logamplitud.

C++ generator of curves in picture a
//#include "uxp.cin" //#include "advacon.cin" 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); int K=200,K1=K+1; DB A=10.; DB dy=2*A/K;  printf("dy=%6.3f",dy); printf("y_0=%6.3f y_K=%6.3f ",Y(0), Y(K));
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x
 * 7) define z_type complex
 * 8) define Re(x) x.real
 * 9) define Im(x) x.imag
 * 10) define I z_type(0.,1.)
 * 1) include "conto.cin"
 * 1) define Y(k) (dy*(k-K/2))

int M=180,M1=M+1; int N=50,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 // z_type tm,tp,F[M1*N1]; z_type tm,tp,F[9681];

//char v[11000]; // v is working array // printf("Output fig01a.eps\n"); FILE *o;o=fopen("analuxp01a.eps","w");ado(o,364,84); fprintf(o,"202 42 translate\n 20 20 scale\n");

//DB sy=4.3/sinh(.04*N/2.); DB sy=2/sinh(.04*N/2.); DO(m,M1) X[m]=-10+.1*(m+.5); DO(n,N1) Y[n]=sy*sinh(.04*(n+.4-N/2));

//for(m=-10;m<9;m++) {M(m,-4)L(m,4)} for(m=-10;m<9;m++) {M(m,-2)L(m,2)} for(n=-2;n<3;n++) {M(-10,n)L(8,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=96;m<106;m++){x=X[m]; //for(m=95;m<106;m++){x=X[m]; for(m=90;m<100;m++){x=X[m]; DO(n,N1){y=Y[n]; z=z_type(x,y); c=z+1.; F[m*N1+n]=c; p=Re(c); q=Im(c); //     if(p>-999 && p<999) g[m*N1+n]=p; //     if(q>-999 && q<999) f[m*N1+n]=q; }               }

for(m=100;m=0;m--) DO(n,N1) {                F[m*N1+n] = log( F[(m+10)*N1+n] ); }

DO(m,M1) DO(n,N1){ c=F[m*N1+n]; p=Re(c); q=Im(c); if(p>-99 && p<99) g[m*N1+n]=p; if(q>-99 && q<99) f[m*N1+n]=q; }

p=1; conto(o,f,w,v,X,Y,M,N, (-3*M_PI ),-5,5); fprintf(o,".04 W 1 0 1 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( -M_PI ),-5,5); fprintf(o,".04 W 1 0 1 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-3.+.1*n),-p, p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (-2. ),-2 ,2); fprintf(o,".03 W 1 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-2.+.1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (-1. ),-2 ,2); fprintf(o,".03 W 1 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-1.+.1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-5,5); fprintf(o,".02 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (   .1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( 1. ),-2 ,2); fprintf(o,".03 W 0 0 1 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, ( 1.+.1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( 2. ),-2 ,2); fprintf(o,".03 W 0 0 1 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, ( 2.+.1*n),-p, p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( M_PI  ),-5,5); fprintf(o,".04 W 1 0 1 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (3*M_PI ),-5,5); fprintf(o,".04 W 1 0 1 RGB S\n");

conto(o,g,w,v,X,Y,M,N, (-4. ),-p,p); fprintf(o,".03 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,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (-2.+.1*n),-p,p); fprintf(o,".01 W 1 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (-1. ),-2,2); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (-1.+.1*n),-p,p); fprintf(o,".01 W 1 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (0. ),-3,3); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (   .1*n),-p,p); fprintf(o,".01 W 0 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N, ( 1. ),-2,2); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, ( 1.+.1*n),-p,p); fprintf(o,".01 W 0 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N, ( 2. ),-p,p); fprintf(o,".03 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, ( 4. ),-p,p); fprintf(o,".03 W 0 0 0 RGB S\n");

for(m=-10;m<9;m++) {M(m,-2.)L(m,2.)} M(-10,0)L(-2,0)fprintf(o,".06 W 1 0 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); //system( "ggv fig01a.eps"); system("epstopdf analuxp01a.eps"); system( "open analuxp01a.pdf"); getchar; system("killall Preview"); }

C++ generator of curves in picture b
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); int K=200,K1=K+1; DB A=10.; DB dy=2*A/K;  printf("dy=%6.3f",dy); printf("y_0=%6.3f y_K=%6.3f ",Y(0), Y(K));
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x
 * 7) define z_type complex
 * 8) define Re(x) x.real
 * 9) define Im(x) x.imag
 * 10) define I z_type(0.,1.)
 * 11) include "analuxpf3c.cin"
 * 12) include "conto.cin"
 * 1) define Y(k) (dy*(k-K/2))

int M=179,M1=M+1; int N=51,N1=N+1; // z_type tm,tp,F[M1*N1]; does not work z_type tm,tp,F[180*52]; 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 // printf("Output fig01b.eps\n"); FILE *o;o=fopen("analuxp01b.eps","w");ado(o,364,84); fprintf(o,"202 42 translate\n 20 20 scale\n");

//DB sy=4.3/sinh(.04*N/2.); DB sy=2/sinh(.04*N/2.); DO(m,M1) X[m]=-10+.1*(m+.5); DO(n,N1) Y[n]=sy*sinh(.04*(n-.4-N/2));

//for(m=-10;m<9;m++) {M(m,-4)L(m,4)} for(m=-10;m<9;m++) {M(m,-2)L(m,2)} for(n=-2;n<3;n++) {M(-10,n)L(8,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=96;m<106;m++){x=X[m]; //for(m=95;m<106;m++){x=X[m]; for(m=90;m<100;m++){x=X[m]; DO(n,N1){y=Y[n]; z=z_type(x,y); //     c=z+1.; c=f3(z); F[m*N1+n]=c; p=Re(c); q=Im(c); //     if(p>-999 && p<999) g[m*N1+n]=p; //     if(q>-999 && q<999) f[m*N1+n]=q; }               }

for(m=100;m=0;m--) DO(n,N1) {                F[m*N1+n] = log( F[(m+10)*N1+n] ); }

DO(m,M1) DO(n,N1){ c=F[m*N1+n]; p=Re(c); q=Im(c); if(p>-999 && p<999) g[m*N1+n]=p; if(q>-999 && q<999) f[m*N1+n]=q; }

p=.8; conto(o,f,w,v,X,Y,M,N, (-3*M_PI ),-4, 4); fprintf(o,".04 W 1 0 1 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( -M_PI ),-4, 4); fprintf(o,".04 W 1 0 1 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-3.+.1*n),-p, p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (-2. ),-2 ,2); fprintf(o,".03 W 1 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-2.+.1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (-1. ),-2 ,2); fprintf(o,".03 W 1 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-1.+.1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-4, 4); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (   .1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( 1. ),-2 ,2); fprintf(o,".03 W 0 0 1 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, ( 1.+.1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( 2. ),-2 ,2); fprintf(o,".03 W 0 0 1 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, ( 2.+.1*n),-p, p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( M_PI  ),-4, 4); fprintf(o,".04 W 1 0 1 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (3*M_PI ),-4, 4); fprintf(o,".04 W 1 0 1 RGB S\n");

conto(o,g,w,v,X,Y,M,N, (-2. ),-p,p); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (-2.+.1*n),-p,p); fprintf(o,".01 W 1 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (-1. ),-1,1); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (-1.+.1*n),-p,p); fprintf(o,".01 W 1 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (0. ),-4,4); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (   .1*n),-p,p); fprintf(o,".01 W 0 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N, ( 1. ),-1,1); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, ( 1.+.1*n),-p,p); fprintf(o,".01 W 0 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N, ( 2. ),-p,p); fprintf(o,".03 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, ( 4. ),-p,p); fprintf(o,".03 W 0 0 0 RGB S\n");

M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf analuxp01b.eps"); system( "open analuxp01b.eps"); getchar; system("killall Preview"); }

C++ generator of curves in picture c
//#include "anluxpf4c.cin" 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); z_type T =2*M_PI/Zo; //z_type r=z_type(1.06,-.96); z_type r=z_type(1.075820830781, - 0.9466419207254); int K=200,K1=K+1; DB A=10.; DB dy=2*A/K;  printf("dy=%6.3f",dy); printf("y_0=%6.3f y_K=%6.3f ",Y(0), Y(K));
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x
 * 7) define z_type complex
 * 8) define Re(x) x.real
 * 9) define Im(x) x.imag
 * 10) define I z_type(0.,1.)
 * 1) include "conto.cin"
 * 1) define Y(k) (dy*(k-K/2))

int M=180,M1=M+1; int N=80,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 //z_type tm,tp,F[M1*N1]; z_type tm,tp,F[181*81];

// printf("Output fig01c.eps\n"); FILE *o;o=fopen("analuxp01c.eps","w");ado(o,364,204); fprintf(o,"202 102 translate\n 20 20 scale\n");

DB sy=4.3/sinh(.04*N/2.); DO(m,M1) X[m]=-10+.1*m; DO(n,N1) Y[n]=sy*sinh(.04*(n-N/2));

for(m=-10;m<9;m++) {M(m,-4)L(m,4)} for(n=-4;n<5;n++) {M( -10,n)L(8,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=5;m<20;m++){int m1; x=X[m]; DO(n,N1){y=Y[n]; z=z_type(x,y);                        //c=F4(z-1.); if(y>.3)       c=Zo+exp(Zo*(z)+r); else if(y<-.3)      c=Zc+exp(Zc*(z)+conj(r)); //z_type(0.318131505204764, 1.337235701430689) else goto ski; // int m1; for(m1=m;m1-999 && p<999) g[m1*N1+n]=p; if(q>-999 && q<999) f[m1*N1+n]=q; c=exp(c); m1+=10; }                               ski:; }               } /* DO(m,M1) DO(n,N1){ c=F[m*N1+n]; p=Re(c); q=Im(c); if(p>-999 && p<999) g[m*N1+n]=p; if(q>-999 && q<999) f[m*N1+n]=q; }

p=2; conto(o,f,w,v,X,Y,M,N, (-3*M_PI ),-999,999); fprintf(o,".04 W 1 0 1 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( -M_PI ),-999,999); fprintf(o,".04 W 1 0 1 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-3.+.1*n),-p, p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (-2. ),-9 ,9); fprintf(o,".03 W 1 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-2.+.1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (-1. ),-9 ,9); fprintf(o,".03 W 1 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-1.+.1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-999,999); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (   .1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( 1. ),-9 ,9); fprintf(o,".03 W 0 0 1 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, ( 1.+.1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( 2. ),-9 ,9); fprintf(o,".03 W 0 0 1 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, ( 2.+.1*n),-p, p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( M_PI  ),-999,999); fprintf(o,".04 W 1 0 1 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (3*M_PI ),-999,999); fprintf(o,".04 W 1 0 1 RGB S\n");

conto(o,g,w,v,X,Y,M,N, (-2. ),-p,p); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (-2.+.1*n),-p,p); fprintf(o,".01 W 1 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (-1. ),-3,3); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (-1.+.1*n),-p,p); fprintf(o,".01 W 1 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (0. ),-999,999); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (   .1*n),-p,p); fprintf(o,".01 W 0 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N, ( 1. ),-3,3); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, ( 1.+.1*n),-p,p); fprintf(o,".01 W 0 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N, ( 2. ),-p,p); fprintf(o,".03 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, ( 4. ),-p,p); fprintf(o,".03 W 0 0 0 RGB S\n");

//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); //system( "ggv fig01c.eps"); system("epstopdf analuxp01c.eps"); system(   "open analuxp01c.eps"); getchar; system("killall Preview"); }

C++ solver of the integral equation with integral Cauchi
z_type Zo=z_type(.31813150520476413, 1.3372357014306895); z_type Zc=z_type(.31813150520476413,-1.3372357014306895);
 * 1) include 
 * 2) include <stdio.h>
 * 3) include <stdlib.h>
 * 4) define DB double
 * 5) include <complex.h>
 * 6) define z_type complex
 * 7) define Re(x) (x).real
 * 8) define Im(x) (x).imag
 * 9) define I z_type(0.,1.)


 * 1) define DO(x,y) for(x=0;x<y;x++)
 * 2) include "f3c.cin"
 * 3) include"ado.cin"
 * 4) define DO(x,y) for(x=0;x<y;x++)
 * 5) define M(x,y) fprintf(o,"%5.3f %5.3f M\n",1.*(x),1.*(y));
 * 6) define L(x,y) fprintf(o,"%5.3f %5.3f L\n",1.*(x),1.*(y));
 * 7) define o(x,y) fprintf(o,"%5.3f %5.3f o\n",1.*(x),1.*(y));

int main{ int j,k,m,n; DB x,y, u, t; z_type z,c,d, cu,cd; int K=NPO; DB A=24.; printf("K=%3d A=%3.1f\n",K,A); int J=K-1; // z_type E[K],F[K],G[K],H[K]; z_type E[2048], F[2048], G[2048], H[2048];
 * 1) include "GLxw2048.inc"

printf( "plot in analuxp0first.eps\n"); FILE *o;o=fopen("analuxp0first.eps","w");ado(o,484,32); fprintf(o,"242 16 translate\n 10 10 scale\n"); for(j=-10;j<11;j+=1){M(j,-1.5)L(j,1.5);} M(-10, 1)L(10 , 1); M(-10 ,-1)L(10 ,-1); fprintf(o,".006 W S\n"); M(-10.1,0)L(10.1,0); fprintf(o,".02 W S\n"); fprintf(o,".01 W S\n 1 setlinejoin\n");

DO(n,K){y=GLx[n]*A; if(y<-3)       E[n]=F[n]=G[n]=Zc; else {if(y>3)  E[n]=F[n]=G[n]=Zo; else { F[n]=c=f3(z_type(0.,y)); E[n]=log(c); G[n]=exp(c); }            }        } DO(k,K){y=GLx[k]*A; u=Re(F[k]);if(k==0)M(y,u)else L(y,u)} DO(k,K){y=GLx[k]*A; u=Im(F[k]);if(k==0)M(y,u)else L(y,u)} fprintf(o,".08 W 1 .4 1 RGB S\n");

//DO(n,K/2) for(n=0;n<K;n+=2) { j=J-n; y=GLx[n]*A; z=z_type(0.,y); c=0.; //printf(" %3d",n); DO(k,K){t=A*GLx[k]; c+= GLw[k]*( G[k]/(z_type( 1.,t)-z) - E[k]/(z_type(-1.,t)-z) );} cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) ); c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd; if(abs(n-K/2)<4)printf("%3d %6.3f %9.6f %9.6f  %9.6f %9.6f\n",n,y,Re(F[n]),Im(F[n]),Re(c),Im(c)); E[n]=log(c);  E[j]=conj(E[n]); F[n]=c;       F[j]=conj(c); G[n]=exp(c);  G[j]=conj(G[n]);}

DO(k,K){y=GLx[k]*A; u=Re(F[k]);if(k==0)M(y,u)else L(y,u)} DO(k,K){y=GLx[k]*A; u=Im(F[k]);if(k==0)M(y,u)else L(y,u)} fprintf(o,".06 W 1 0 0 RGB S\n");

DO(n,K) H[n]=F[n]; //DO(n,K/2) for(n=1;n<K;n+=2) { j=J-n; y=GLx[n]*A; z=z_type(0.,y); c=0.; //printf(" %3d",n); DO(k,K){t=A*GLx[k]; c+= GLw[k]*( G[k]/(z_type( 1.,t)-z) - E[k]/(z_type(-1.,t)-z) );} cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) ); c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd; if(abs(n-K/2)<4)printf("%3d %6.3f %9.6f %9.6f  %9.6f %9.6f\n",n,y,Re(F[n]),Im(F[n]),Re(c),Im(c)); E[n]=log(c);  E[j]=conj(E[n]); F[n]=c;       F[j]=conj(c); G[n]=exp(c);  G[j]=conj(G[n]);}

DO(k,K){y=GLx[k]*A; u=Re(F[k]);if(k==0)M(y,u)else L(y,u)} DO(k,K){y=GLx[k]*A; u=Im(F[k]);if(k==0)M(y,u)else L(y,u)} fprintf(o,".04 W 0 1 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); //system( "ggv fig0first.eps"); system("epstopdf analuxp0first.eps"); system(   "open analuxp0first.pdf");

printf("plot in analuxp0test.eps\n"); o=fopen("analuxp0test.eps","w");ado(o,484,52); fprintf(o,"242 16 translate\n 10 10 scale\n"); for(j=-10;j<11;j+=1){M(j,-1.5)L(j,1.5);} M(-10, 1)L(10 , 1); M(-10 ,-1)L(10 ,-1); fprintf(o,".006 W S\n"); M(-10.1,0)L(10.1,0); fprintf(o,".01 W S\n 1 setlinejoin\n");

DO(k,K){y=GLx[k]*A; u=100*Re(H[k]-F[k]);if(k==0)M(y,u)else L(y,u)} DO(k,K){y=GLx[k]*A; u=100*Im(H[k]-F[k]);if(k==0)M(y,u)else L(y,u)} fprintf(o,".04 W 1 0 0 RGB S\n");

DO(n,K) H[n]=F[n]; DO(n,K){ j=J-n; y=GLx[n]*A; z=z_type(0.,y); c=0.; //printf(" %3d",n); DO(k,K){t=A*GLx[k]; c+= GLw[k]*( G[k]/(z_type( 1.,t)-z) - E[k]/(z_type(-1.,t)-z) );} cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) ); c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd; if(abs(n-K/2)<4)printf("%3d %6.3f %9.6f %9.6f  %9.6f %9.6f\n",n,y,Re(F[n]),Im(F[n]),Re(c),Im(c)); E[n]=log(c);  E[j]=conj(E[n]); F[n]=c;       F[j]=conj(c); G[n]=exp(c);  G[j]=conj(G[n]);}

DO(k,K){y=GLx[k]*A; u=100*Re(H[k]-F[k]);if(k==0)M(y,u)else L(y,u)} DO(k,K){y=GLx[k]*A; u=100*Im(H[k]-F[k]);if(k==0)M(y,u)else L(y,u)} fprintf(o,".03 W 0 1 0 RGB S\n");

DO(n,K) H[n]=F[n]; DO(n,K){ j=J-n; y=GLx[n]*A; z=z_type(0.,y); c=0.; //printf(" %3d",n); DO(k,K){t=A*GLx[k]; c+= GLw[k]*( G[k]/(z_type( 1.,t)-z) - E[k]/(z_type(-1.,t)-z) );} cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) ); c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd; if(abs(n-K/2)<4)printf("%3d %6.3f %9.6f %9.6f  %9.6f %9.6f\n",n,y,Re(F[n]),Im(F[n]),Re(c),Im(c)); E[n]=log(c);  E[j]=conj(E[n]); F[n]=c;       F[j]=conj(c); G[n]=exp(c);  G[j]=conj(G[n]);}

DO(k,K){y=GLx[k]*A; u=100*Re(H[k]-F[k]);if(k==0)M(y,u)else L(y,u)} DO(k,K){y=GLx[k]*A; u=100*Im(H[k]-F[k]);if(k==0)M(y,u)else L(y,u)} fprintf(o,".02 W 0 0 1 RGB S\n");

DO(n,K) H[n]=F[n]; DO(m,64) for(n=0;n<K;n+=2) { j=J-n; y=GLx[n]*A; z=z_type(0.,y); c=0.; //printf(" %3d",n); DO(k,K){t=A*GLx[k]; c+= GLw[k]*( G[k]/(z_type( 1.,t)-z) - E[k]/(z_type(-1.,t)-z) );} cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) ); c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd; if(n==K/2)printf("%3d %17.13f %17.14f %17.14f %17.14f\n",m,Re(F[n]),Im(F[n]),Re(c),Im(c)); E[n]=log(c);  E[j]=conj(E[n]); F[n]=c;       F[j]=conj(c); G[n]=exp(c);  G[j]=conj(G[n]);}

DO(k,K){y=GLx[k]*A; u=100*Re(H[k]-F[k]);if(k==0)M(y,u)else L(y,u)} DO(k,K){y=GLx[k]*A; u=100*Im(H[k]-F[k]);if(k==0)M(y,u)else L(y,u)} fprintf(o,".02 W .4 0 .4 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); //system( "ggv fig0test.eps"); system("epstopdf analuxp0test.eps"); system( "open analuxp0test.pdf");

o=fopen("analuxp0.dat","w"); DO(k,K) fprintf(o,"%4d %18.14f %18.14f\n",k,Re(F[k]),Im(F[k])); fclose(o);

getchar; system("killall Preview"); }

C++ generator of the input file
int main{ int k,n; DB x,y; FILE *oo,*o; int K=2048; DB A=24.; printf("K=%3d A=%3.1f\n",K,A); oo=fopen("analuxp0.dat","r"); o=fopen("analuxpf2048.inc","w"); fprintf(o,"DB A=%8.4f;\n",A); fprintf(o,"int K=%4d;\n",K); fprintf(o,"// perhaps, A=24, and NPO =2048; NPO is supposed to be defined in GLxw.\n"); fprintf(o,"z_type F[%4d]={\n",K); //fscanf(oo,"%d%lf%lf",&n,&x,&y); printf("%5d %5d %17.14f %17.14f \n",n,x,y); DO(k,K) {       fscanf(oo,"%d%lf%lf",&n,&x,&y);if(k!=n){printf("%5d %5d %17.14f %17.14f ?\n",k,n,x,y);} fprintf(o,"z_type(%16.14f,%16.14f)",x,y); if(k<K-1) fprintf(o,",\n"); else fprintf(o,"};\n"); } fclose(oo); fclose(o); printf("read from analuxp0.dat write  analuxpf2048.inc\n"); printf("Perhaps, you need also analuxp10x1.cin \n"); }
 * 1) include <math.h>
 * 2) include <stdio.h>
 * 3) include <stdlib.h>
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)

C++ generator of curves in picture d
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); int K=200,K1=K+1; DB A=10.; DB dy=2*A/K;  printf("dy=%6.3f",dy); printf("y_0=%6.3f y_K=%6.3f ",Y(0), Y(K));
 * 1) include <math.h>
 * 2) include <stdio.h>
 * 3) include <stdlib.h>
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include <complex.h>
 * 7) define z_type complex
 * 8) define Re(x) x.real
 * 9) define Im(x) x.imag
 * 10) define I z_type(0.,1.)
 * 11) include "analuxpf4c.cin"
 * 12) include "conto.cin"
 * 1) define Y(k) (dy*(k-K/2))

int M=180,M1=M+1; int N=80,N1=N+1; DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array. //z_type tm,tp,F[M1*N1]; z_type tm,tp,F[181*81]; char v[M1*N1]; // v is working array // printf("Output fig01d.eps\n"); FILE *o;o=fopen("analuxp01d.eps","w");ado(o,364,204); fprintf(o,"202 102 translate\n 20 20 scale\n");

DB sy=4.3/sinh(.04*N/2.); DO(m,M1) X[m]=-10+.1*m; DO(n,N1) Y[n]=sy*sinh(.04*(n-N/2));

//for(m=-10;m<9;m++) {M(m,-4)L(m,4)} for(m=-10;m<9;m++) {M(m,-4)L(m,4)} for(n=-4;n<5;n++) {M( -10,n)L(8,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=96;m<106;m++){x=X[m]; for(m=95;m<106;m++){x=X[m]; DO(n,N1){y=Y[n]; z=z_type(x,y); c=F4(z); F[m*N1+n]=c; p=Re(c); q=Im(c); if(p>-999 && p<999) g[m*N1+n]=p; if(q>-999 && q<999) f[m*N1+n]=q; }               }

for(m=106;m<M1;m++) DO(n,N1) {                F[m*N1+n] = exp( F[(m-10)*N1+n] ); } for(m=94;m>=0;m--) DO(n,N1) {                F[m*N1+n] = log( F[(m+10)*N1+n] ); }

DO(m,M1) DO(n,N1){ c=F[m*N1+n]; p=Re(c); q=Im(c); if(p>-999 && p<999) g[m*N1+n]=p; if(q>-999 && q<999) f[m*N1+n]=q; }

p=2; conto(o,f,w,v,X,Y,M,N, (-3*M_PI ),-999,999); fprintf(o,".04 W 1 0 1 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( -M_PI ),-999,999); fprintf(o,".04 W 1 0 1 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-3.+.1*n),-p, p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (-2. ),-9 ,9); fprintf(o,".03 W 1 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-2.+.1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (-1. ),-9 ,9); fprintf(o,".03 W 1 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-1.+.1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-999,999); fprintf(o,".02 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (   .1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( 1. ),-9 ,9); fprintf(o,".03 W 0 0 1 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, ( 1.+.1*n),-p ,p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( 2. ),-9 ,9); fprintf(o,".03 W 0 0 1 RGB S\n"); for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, ( 2.+.1*n),-p, p); fprintf(o,".01 W 0 .6 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, ( M_PI  ),-99,99); fprintf(o,".04 W 1 0 1 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (3*M_PI ),-99,99); fprintf(o,".04 W 1 0 1 RGB S\n");

conto(o,g,w,v,X,Y,M,N, (-2. ),-p,p); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (-2.+.1*n),-p,p); fprintf(o,".01 W 1 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (-1. ),-3,3); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (-1.+.1*n),-p,p); fprintf(o,".01 W 1 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (0. ),-99,99); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (   .1*n),-p,p); fprintf(o,".01 W 0 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N, ( 1. ),-3,3); fprintf(o,".03 W 0 0 0 RGB S\n"); for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, ( 1.+.1*n),-p,p); fprintf(o,".01 W 0 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N, ( 2. ),-p,p); fprintf(o,".03 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, ( 4. ),-p,p); fprintf(o,".03 W 0 0 0 RGB S\n");

M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); //system( "ggv fig01d.eps"); system("epstopdf analuxp01d.eps"); system(   "open analuxp01d.eps"); getchar; system("killall Preview"); }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphics} \usepackage{rotating} \paperwidth 375px \paperheight 542px \textwidth 500pt \textheight 900pt \topmargin -100pt \oddsidemargin -66pt \parindent 0pt \pagestyle{empty} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document} \sx{1.}{\begin{picture}(370,80) %\put(5,5){\includegraphics{fig01a}} \put(6,6){\includegraphics{analuxp01a}} \put(12,60){\sx{3}{a}} \put( 0,82){\sx{1.2}{$y$}} \put( 0,63){\sx{1.1}{$1$}} \put( 0,43){\sx{1.1}{$0$}} \put(-9,23){\sx{1.1}{$-1$}} \put( 18,-2){\sx{1.1}{$-8$}} \put( 58,-2){\sx{1.1}{$-6$}} \put(98,-2){\sx{1.1}{$-4$}} \put(138,-2){\sx{1.1}{$-2$}} \put(186,-2){\sx{1.1}{$0$}} \put(226,-2){\sx{1.1}{$2$}} \put(266,-2){\sx{1.1}{$4$}} \put(306,-2){\sx{1.1}{$6$}} \put(346,-2){\sx{1.1}{$8$}} \put(362,-2){\sx{1.2}{$x$}} \put(95,52){\sx{.8}{$v\!=\!1$}} \put(177,57){\sx{.8}{\rot{58}$v\!=\!1$\ero}} \put(172,24.2){\sx{.8}{\rot{32}{$u\!=\!0$}\ero}} \put(190,66){\sx{.8}{$v\!=\!1$}} \put(180,45.8){\sx{.8}{$v\!=\!0$}} \end{picture}}

\sx{1.}{\begin{picture}(370,96) \put( 6,6){\includegraphics{analuxp01b}} \put(12,60){\sx{3}{b}} \put( 0,82){\sx{1.2}{$y$}} \put( 0,63){\sx{1.1}{$1$}} \put( 0,43){\sx{1.1}{$0$}} \put(-9,23){\sx{1.1}{$-1$}} \put( 18,-2){\sx{1.1}{$-8$}} \put( 58,-2){\sx{1.1}{$-6$}} \put( 98,-2){\sx{1.1}{$-4$}} \put(138,-2){\sx{1.1}{$-2$}} \put(186,-2){\sx{1.1}{$0$}} \put(226,-2){\sx{1.1}{$2$}} \put(266,-2){\sx{1.1}{$4$}} \put(306,-2){\sx{1.1}{$6$}} \put(346,-2){\sx{1.1}{$8$}} \put(362,-2){\sx{1.2}{$x$}} \multiput(40,66)(90,21){2}{\sx{.8}{$v\!=\!1.4$}} \multiput(93,77)(-90,-21){1}{\sx{.8}{$u\!=\!1.2$}} \multiput(183,83)(-90,-21){2}{\sx{.8}{$v\!=\!1.2$}} \multiput(185,73)(-90,-21){2}{\sx{.8}{$v\!=\!1$}} \put(184,46){\sx{.8}{$v\!=\!0$}} \put(184,20){\sx{.8}{$v\!=\!-1$}} \put(176,21){\sx{.8}{\rot{42}{$u\!=\!0$}\ero}} \put(211,39){\sx{.8}{\rot{90}{$u\!=\!1$}\ero}} \multiput(4,34)(90,-21){2}{\sx{.8}{$u\!=\!0.4$}} \multiput(38,26)(90,-21){1}{\sx{.8}{$v\!=\!-1.4$}} \end{picture}}

\sx{1.}{\begin{picture}(366,176) \put( 6,-14){\includegraphics{analuxp01c}} \put(35,153){\sx{1.}{$u\!+\!\mathrm i v \approx L\approx 0.3+1.3\mathrm i$}} \put(33,13){\sx{1.}{$u\!+\!\mathrm i v \approx L^*\approx 0.3-1.3\mathrm i$}} \put(12,140){\sx{3}{c}} \put( 0,164){\sx{1.2}{$y$}} \put( 0,144){\sx{1.1}{$3$}} \put( 0,124){\sx{1.1}{$2$}} \put( 0,104){\sx{1.1}{$1$}} \put( 0, 84){\sx{1.1}{$0$}} \put(-9, 64){\sx{1.1}{$-1$}} \put(-9, 44){\sx{1.1}{$-2$}} \put(-9, 24){\sx{1.1}{$-3$}} \put( 18,-2){\sx{1.1}{$-8$}} \put( 58,-2){\sx{1.1}{$-6$}} \put( 98,-2){\sx{1.1}{$-4$}} \put(138,-2){\sx{1.1}{$-2$}} \put(186,-2){\sx{1.1}{$0$}} \put(226,-2){\sx{1.1}{$2$}} \put(266,-2){\sx{1.1}{$4$}} \put(306,-2){\sx{1.1}{$6$}} \put(346,-2){\sx{1.1}{$8$}} \put(362,-2){\sx{1.2}{$x$}} \multiput(271,159)(-90,-21){4}{\sx{.8}{$u\!=\!0.4$}} \multiput(271,147)(-90,-21){3}{\sx{.8}{$v\!=\!1.2$}} \multiput(185,113)(90,21){2}{\sx{.8}{$v\!=\!1$}} %\put(184,86){\sx{.8}{$v\!=\!0$}} \multiput(184,60)(90,-21){2}{\sx{.8}{$v\!=\!-1$}} \multiput(176,61)(89.4,-21){3}{\sx{.8}{\rot{42}{$u\!=\!0$}\ero}}

\put(211,79){\sx{.8}{\rot{90}{$u\!=\!1$}\ero}} \multiput(40,106)(90,21){4}{\sx{.8}{$v\!=\!1.4$}} \multiput(40,66)(90,-21){4}{\sx{.8}{$v\!=\!-1.4$}} \multiput(5,75)(90,-21){4}{\sx{.8}{$u\!=\!0.4$}} \end{picture}}

\sx{1.}{\begin{picture}(366,176) \put( 6,-14){\includegraphics{analuxp01d}} \put(35,153){\sx{1.}{$u\!+\!\mathrm i v \approx L\approx 0.3+1.3\mathrm i$}} \put(33,13){\sx{1.}{$u\!+\!\mathrm i v \approx L^*\approx 0.3-1.3\mathrm i$}} \put(12,140){\sx{3}{d}} \put( 0,162){\sx{1.2}{$y$}} \put( 0,144){\sx{1.1}{$3$}} \put( 0,124){\sx{1.1}{$2$}} \put( 0,104){\sx{1.1}{$1$}} \put( 0, 84){\sx{1.1}{$0$}} \put(-9, 64){\sx{1.1}{$-1$}} \put(-9, 44){\sx{1.1}{$-2$}} \put(-9, 24){\sx{1.1}{$-3$}} \put( 18,-2){\sx{1.1}{$-8$}} \put( 58,-2){\sx{1.1}{$-6$}} \put( 98,-2){\sx{1.1}{$-4$}} \put(138,-2){\sx{1.1}{$-2$}} \put(186,-2){\sx{1.1}{$0$}} \put(226,-2){\sx{1.1}{$2$}} \put(266,-2){\sx{1.1}{$4$}} \put(306,-2){\sx{1.1}{$6$}} \put(346,-2){\sx{1.1}{$8$}} \put(362,-2){\sx{1.2}{$x$}} \multiput(271,159)(-90,-21){4}{\sx{.8}{$u\!=\!0.4$}} \multiput(271,147)(-90,-21){3}{\sx{.8}{$v\!=\!1.2$}} \multiput(185,113)(90,21){2}{\sx{.8}{$v\!=\!1$}} \put(184,86){\sx{.8}{$v\!=\!0$}} \multiput(184,60)(90,-21){2}{\sx{.8}{$v\!=\!-1$}} \multiput(176,61)(89.5,-21){3}{\sx{.8}{\rot{42}{$u\!=\!0$}\ero}} \put(211,79){\sx{.8}{\rot{90}{$u\!=\!1$}\ero}} \multiput(40,106)(90,21){4}{\sx{.8}{$v\!=\!1.4$}} \multiput(40,66)(90,-21){4}{\sx{.8}{$v\!=\!-1.4$}} \multiput(5,75)(90,-21){4}{\sx{.8}{$u\!=\!0.4$}}

\end{picture}} \end{document}