I have to write a code in Matlab that explains a conformal mapping under the transformation w=z^2+1/z. How can I do that?(I've already read the link: Exploring a Conformal Mapping, but with this particular transformation I failed).

Answer by Anton Semechko
on 9 Jul 2018

Edited by Anton Semechko
on 9 Jul 2018

Modified code ('conformal_map_demo') is attached below. In principle, this piece of code should should allow you to visualize any complex analytic function. You just have to modify the 'f' function manually.

Example for f=z^2

Example for f=z^2+1/z

function conformal_map_demo

% Grid settings

XLim=5*[-1 1];

n = 20;

dX=(XLim(2)-XLim(1))/n;

% Initialize figure

figure

ha1=subplot(1, 2, 1);

title ('Grid of Squares')

axis('equal')

hold on

ha2=subplot(1, 2, 2);

title ('Image Of Grid Under w = z^2 + 1/z')

axis('equal')

hold on

% ============================== PRE-IMAGE ================================

axes(ha1)

% Draw reference square at top-right corner

% -------------------------------------------------------------------------

su=[0 0; 1 0; 1 1; 0 1]; % unit square

s=bsxfun(@plus,dX*su,XLim(2)*[1 1]-dX);

fill(s(:,1),s(:,2),[0.9 0.9 0.9])

% Draw vertical grid lines at dX intervals

% -------------------------------------------------------------------------

for x=XLim(1):dX:XLim(2)

plot(x*ones(1,2),XLim,'b')

end

% Draw horizontal grid lines at dX intervals

% -------------------------------------------------------------------------

for y=XLim(1):dX:XLim(2)

plot(XLim,y*ones(1,2), 'r')

end

% Draw Unit Tangents for the reference square

% -------------------------------------------------------------------------

x1 = XLim(2)-dX;

y1 = x1;

% 1. Draw the Unit Tangent in the i-direction

a = [x1 x1];

b = y1 + dX*[0 1];

line(a, b, 'linewidth',3, 'color', 'blue')

% 2. Draw the Unit Tangent in the r-direction

a = x1 + dX*[0 1];

b = [y1 y1];

line(a,b, 'linewidth',3, 'color', 'red')

hold off

% Set axes domain, and range

axis((XLim(2)+dX)*[-1 1 -1 1])

% ================================ IMAGE ==================================

axes(ha2)

f=@(z) z.^2 + 1./z;

% Draw the image of the reference square

% -------------------------------------------------------------------------

% Subdivide original reference square; to insert more points between corners

for i=1:8

s_new=(s+circshift(s,[-1 0]))/2;

s=reshape(cat(1,s',s_new'),2,[]);

s=s';

end

f_s=f(s(:,1)+1i*s(:,2));

fill(real(f_s),imag(f_s),[0.9 0.9 0.9])

% Draw images of the vertical lines

% -------------------------------------------------------------------------

BB=Inf*[1 -1;1 -1];

yy=linspace(XLim(1),XLim(2),1E3)';

for x = XLim(1):dX:XLim(2);

w=f(x*ones(1E3,1) + 1i*yy);

u=real(w);

v=imag(w);

plot(u,v,'-b')

if x~=0

BB(:,1)=min(BB(:,1),[min(u);min(v)]);

BB(:,2)=max(BB(:,2),[max(u);max(v)]);

end

end

% Draw the images of the horizontal lines

% -------------------------------------------------------------------------

xx=yy;

for y = XLim(1):dX:XLim(2);

w=f(xx+1i*y*ones(1E3,1));

u=real(w);

v=imag(w);

plot(u,v,'-r')

if y~=0

BB(:,1)=min(BB(:,1),[min(u);min(v)]);

BB(:,2)=max(BB(:,2),[max(u);max(v)]);

end

end

% Draw the images of the unit tangents under w

% -------------------------------------------------------------------------

% Jacobian the map

syms x y

Jxx=diff(real(f(x+1i*y)),'x');

Jxy=diff(real(f(x+1i*y)),'y');

Jyy=diff(imag(f(x+1i*y)),'y');

Jyx=diff(imag(f(x+1i*y)),'x');

J=[Jxx Jxy; Jyx Jyy];

% Evaluate Jacobian at the bottom-left corner of the reference square

c=XLim(2)*[1 1];

Jc=dX*double(subs(J,[x,y],c));

% Visualize new tangent vectors at c

[Ux,Uy]=deal([real(f_s(1)) imag(f_s(1))]);

Ux=[Ux;Ux+Jc(:,1)'];

Uy=[Uy;Uy+Jc(:,2)'];

plot(Ux(:,1),Ux(:,2),'-r','LineWidth',3)

plot(Uy(:,1),Uy(:,2),'-b','LineWidth',3)

% Set axes domain and range

BB(:,1)=min(BB(:,1),min([Ux;Uy])');

BB(:,2)=max(BB(:,2),max([Ux;Uy])');

dB=BB(:,2)-BB(:,1);

BB(:,1)=BB(:,1)-0.025*dB;

BB(:,2)=BB(:,2)+0.025*dB;

set(ha2,'XLim',BB(1,:),'YLim',BB(2,:))

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