Transformation of the complex plan: Conformal Mapping

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).

[Anton Semechko]

Post your code on here to show us what you have done so far and explain where the code fails

[Fe21]

I tried to follow steps of the link, but steps from 3 to 8 need to reverse the transformation and my transformation is difficult to reverse

[Fe21]

And so how can I proceed?

[Anton Semechko]

Proceed with what?

[Fe21]

You said that my transformation is not conformal, so I guess I can not use the steps...

[Anton Semechko]

I have no idea what steps you are talking about because you haven't posted any code or a link to the reference you are using

[Fe21]

Sorry! Here it is: https://it.mathworks.com/help/images/examples/exploring-a-conformal-mapping.html

[Anton Semechko]

Transformation being used in that example is

(z + 1/z)/2

and not

 z^2 + 1/z

You can find more info about, and examples of, conformal transformations in this reference

[Fe21]

so,if I understand,I can not proceed with the methods seen in the link

[Anton Semechko]

I was wrong. Mobius transformations are a group of conformal transformations of a Riemann sphere. Complex plane admits a much larger group of comformal transformations. According to the reference link I posted, any complex analytic function with non-zero first derivative is also a conformal map (pp. 34). So the function w=z^2 + 1/z is a conformal map. To find its inverse you will have to solve z^3 - w*z + 1 = 0 (e.g., using 'roots' function). Unfortunately, resulting expression is a "bit" more complicated compared to the one for z^2 - 2*w*z + 1 = 0. Here it is:

syms w
r=roots([1 0 w 1]);
pretty(simplify(r))     

/                  w           \
|           #1 - ----          |
|                3 #1          |
|                              |
|   #3 + sqrt(3) w 1i + #2 - w |
| - -------------------------- |
|              6 #1            |
|                              |
|  w + sqrt(3) w 1i + #2 - #3  |
|  --------------------------  |
\             6 #1             /

where

           /     /  3     \     \1/3
           |     | w    1 |   1 |
     #1 == | sqrt| -- + - | - - |
           \     \ 27   4 /   2 /

                   /     /  3     \     \2/3
                   |     | w    1 |   1 |
     #2 == sqrt(3) | sqrt| -- + - | - - |    3i
                   \     \ 27   4 /   2 /

             /     /  3     \     \2/3
             |     | w    1 |   1 |
     #3 == 3 | sqrt| -- + - | - - |
             \     \ 27   4 /   2 /

Keep in mind that "sqrt" term inside #1, #2, and #3 can have both +ve and -ve signs.

[Fe21]

I do not think I have the necessary requirements to understand the whole file. Then you told me that mine is not a conformal transformation, so maybe I have to change method.

I have an example of code that does the same thing, but with the transformation w=z^2. How can I adapt it for my transformation?

[Anton Semechko]

Sorry about the confusion. Give me a few minutes to modify your code.

Answer 1

[Anton Semechko]

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,:))

Answer 2

[abd abd1]

here is my code in MATLAB to generate julia set in complex domain

Answer 3

[Hamed Najafi]

This code might also be helpful:

https://www.mathworks.com/matlabcentral/fileexchange/160521-conformal-mapping