Can two conformal mappings be combined?

A conformal mapping function within a complex domain maps the unit circle in the ζ-plane to the -plane. Similarly, there is another function . Combining and , we can obtain .

Can this function also satisfy conformal mapping, and are there any applicable conditions for it to do so?

The following figure is an attempt I made. If feasible, can any smooth and simply connected region be combined with for conformal mapping?

Answer 1

[Jaskirat]

Hello!

I understand that the objective in question is to check whether given functions are conformal and if the conformal mapping can be done on MATLAB.

A function is conformal if it is holomorphic, that is, if it is complex differentiable at each point in its domain. This can be checked by using the "Cauchy-Riemann" equations on the partial derivatives of the function. The code for this is given below.

syms x y real
u = % define the real part of the function
v = % define the imaginary part of the function

% Compute partial derivatives
ux = diff(u, x);
uy = diff(u, y);
vx = diff(v, x);
vy = diff(v, y);

% Check Cauchy-Riemann equations
isHolomorphic = simplify(ux - vy) == 0 && simplify(uy + vx) == 0;

if isHolomorphic
    disp('The function is holomorphic.');
else
    disp('The function is not holomorphic.');
end

Creating a conformal mapping in MATLAB requires creating the function for mapping over the shared domain and using the “subplot” function to plot the mapping.

An alternative approach involves utilizing an external package known as "Chebfun," which offers a suite of tools for function manipulation, including conformal mapping capabilities. This package can be conveniently installed via MATLAB's "Add-On Explorer."

Refer the below file exchange link for more information on the package.

https://www.mathworks.com/matlabcentral/fileexchange/47023-chebfun-current-version#functions_tab

Hope this helps!

[Shengfan Bi]

Thank you for your detailed response. Using the concept of holomorphicity to determine if a function is conformal is an excellent approach, and the plugin works very well.