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E1P1
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%(This code can be directly copy pasted into Matlab.)
%We make a grid of points in the unit disc
%and store the points in a matrix z.
N=100;e=0.001;
r=linspace(0,1-e,N);t=linspace(0,1,N)*2*pi;
z=r'*exp(1i*t);
%The next step will make the grid more dense 
%near a point on the boundary. Here we produce
%a more dense grid near points 1 and -1.
k=100;l=1:k;t=2.^l;T=[-t,t];[x,y]=meshgrid(t,T);a=(x+1i*y);b=(a-1)./(a+1);
z=[z;b;-b];
%Compare this to the following:
%z=[z;b];
%theta=pi/2;[z;exp(1i*theta)*b];

%Now, we open a new figure window and draw the first picture there.
figure('units','normalized','outerposition',[0 0 1 1])
subplot(1,2,1)
plot(z,'.b')
%We add a title for the first picture.
title('$z\in\textbf{D}$','interpreter','latex','fontsize',20)
%We adjust the axes and add a grid.
axis equal	
K=2;
axis([-K,K,-K,K])
grid on
%We plot the second picture.
subplot(1,2,2)
w=z-z.^2/2;
plot(w,'.r')
axis equal
title('$w=z-\frac{1}{2}z^2$','interpreter','latex','fontsize',20)
axis equal
axis([-K,K,-K,K])
grid on

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E1P2
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%We make a grid of points in the unit disc.
N=100;e=0.001;
r=linspace(0,1-e,N);t=linspace(0,1,N)*2*pi;
z=r'*exp(1i*t);
k=100;l=1:k;t=2.^l;T=[-t,t];[x,y]=meshgrid(t,T);a=(x+1i*y);b=(a-1)./(a+1);
z=[z;b;-b];
%We draw the unit disc,
figure('units','normalized','outerposition',[0 0 1 1])
subplot(2,2,1)	%2x2 plots, this is the first one
plot(z,'.b')	
title('$z\in\textbf{D}$','interpreter','latex','fontsize',20)
axis equal
%half plane,
subplot(2,2,2)	%second plot
w=(1+z)./(1-z);	%calculate w
plot(w,'.r')	%plot w
L=10;		%adjust the limits of the image area
axis([-L,L,-L,L])
title('$w=\frac{1+z}{1-z}$','interpreter','latex','fontsize',20)
%sector,
subplot(2,2,3)
pow=1/2;	%now the power is 1/2. you can change the power (*)
q=w.^(pow);	
plot(q,'.g')
axis([-L,L,-L,L])
title('$w^{1/2}$','interpreter','latex','fontsize',20)
%and the lens with unit disc on the back.
subplot(2,2,4)
c=(q-1)./(q+1);
plot(z,'.','color',[1 0.5 0.1])
hold on		
plot(c,'.k')
axis equal
title('$f(z)=\frac{w^{1/2}-1}{w^{1/2}+1}$','interpreter',...
'latex','fontsize',20)