How to apply quadgk in solving this integral?

Here, z is a complex variable such that z belongs to unit disk,i.e., . is an analytic function. is also an analytic function such that .

How can we apply Gauss konrod quadrature on a function having two variables like above? i'm attaching a hint if you understand.

i think they're making a mesh for values of z. But i am not able to understand how. Also, they have plotted the images of unit disk under the maps and using quadgk in the following example. if someone could help me with the code to draw like these:

Please help me understand how they did this and help me with the code.

[Star Strider]

Proviide the appropriate code forφ and ω and we will see what we can do.

[simran]

and where is the appell hypergeometric function.

[Star Strider]

We will wait for you to code those and post the code.

[simran]

What? Didn't you understand what have I asked?

Above are the screenshots of a concept which I didn't understand. How have they applied gauss konrod quadrature method on a two variable function in z and t. How have they made mesh of z, and plotted the images of $\phi$ and $\f(z)$ by integrating the R.H.S of f(z) function.

[Star Strider]

We’ll help you with the integration (since that may not be straightforward, especially if waypoints are involved, the functions are complex so they requires complex integration techniques, and other complicating factors), however you need to code the functions you want to integrate.

So code them, try the integration, and if you have problems, describe them with the code and any error messages that are thrown in the process. If there are no error messages and the code didn’t do what you wanted, or did something you didn’t want, describe that as well.

And of course, show all the code you use, and attach any data files necessary to run it, and how you use the variables read from them.

[simran]

If I knew it already, why would I be asking? And this is not done by complex integration techniques. They have just applied gauss kronrod quadrature. Even I didn't understand how they did it.

[Torsten]

Example for phi(z) = -cos(z), omega(z) = z^4:

z = 0.9*(1 + 2*1i)/sqrt(5);
phidot = @(t) sin(z*t);
omega = @(t) (z*t).^4;
fun = @(t) phidot(t)./(1-omega(t)) * z;
f = 2*real(integral(fun,0,1)) - (conj(-cos(z)))

f = 0.9239 + 0.3505i

[Star Strider]

It looks like hypergeom could be the correct function, so start there with the Symbolic Math Toolbox to create φ and ω. You may also need to use the diff function. If is the complex conjugate, there are functions for that as well.

Integrating it numerically will likely require that you then use the matlabFunction function with your symbolically-coded integrand to create an anonymous function version of it that you can then use with any of the numerical integration functions to do the actual integration (assuming that you cannot integrate it symbolically).

[Sam Chak]

Just to provide a brief update, I believe these spider-web-like plots are referred to as 'Conformal Mapping,' as mentioned on Wolfram MathWorld. The screenshots are excerpts from the article 'Harmonic Shears and Numerical Conformal Mappings,' authored by Tri Quach and published in the journal Filomat 31:8 (2017), pages 2231–2241.

OP @simran is interested in learning how to numerically plot these Conformal Maps using the Gauss-Kronrod quadrature method, which is also available as a built-in function in MATLAB called 'quadgk()'. Approximately 10 days ago, @Angelo Yeo plotted something similar on this page:

https://www.mathworks.com/matlabcentral/answers/2012102-how-do-i-plot-this-in-matlab?s_tid=srchtitle

n_rays = 16 + 1;
n_circles = 9 + 1;

n_samples = 200;
rays = linspace(0, 1, n_samples)' * exp(1j*linspace(0, 2*pi, n_rays));
interior_circles = linspace(0, 1, n_circles)' * (cos(linspace(0, 2*pi, n_samples)) + 1j * sin(linspace(0, 2*pi, n_samples)));
interior_circles = transpose(interior_circles);

%% plotting
figure;
subplot(1,2,1);
for i_rays = 1:n_rays
    plot(real(rays(:,i_rays)), imag(rays(:,i_rays)),'Color',[88, 140, 102]/ 255)
    hold on;
end

for i_circle = 1:n_circles
    plot(real(interior_circles(:,i_circle)), imag(interior_circles(:,i_circle)),'Color','r')
end
axis square
xlim([-2, 2])
ylim([-2, 2])
grid on;

subplot(1,2,2);
f = @(z) z./(1-z); % complex function to apply
for i_rays = 1:n_rays
    plot(real(f(rays(:,i_rays))), imag(f(rays(:,i_rays))),'Color',[88, 140, 102]/ 255)
    hold on;
end

for i_circle = 1:n_circles
    plot(real(f(interior_circles(:,i_circle))), imag(f(interior_circles(:,i_circle))),'Color','r')
end
axis square
xlim([-2, 2])
ylim([-2, 2])
grid on;

[Sam Chak]

According to Tri Quach, the Schwarz–Christoffel Toolbox (developed by Prof. Toby Driscoll) is used to provide conformal maps.

https://tobydriscoll.net/project/sc-toolbox/

[simran]

@Sam Chak yes you understood it right. But only $\phi$ is the conformal map for which I guess they used SC toolbox. $f(z)$ is a harmonic mapping (may not be conformal) plotted with the help of algorithm they presented by using quadgk.

Yes I am trying to learn how to plot these numerically using quadgk.