How to apply quadgk in solving this integral?

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simran 2023-9-2

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2016061-how-to-apply-quadgk-in-solving-this-integral

编辑： simran 2023-9-3

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.

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Sam Chak 2023-9-2

在 MATLAB Online 中打开

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 2023-9-2

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 2023-9-3

编辑：simran 2023-9-3

@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.

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MATLAB Graphics 2-D and 3-D Plots Surfaces, Volumes, and Polygons Surface and Mesh Plots

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