To plot a complete Riemann surface for the function
, you need to consider the nature of the complex square root function and how it behaves in different branches of the complex plane. The function 
has two branches, which you've correctly identified. However, to visualize the complete surface, you must ensure that your parameterization covers the entire complex plane, not just a portion.
, you need to consider the nature of the complex square root function and how it behaves in different branches of the complex plane. The function has two branches, which you've correctly identified. However, to visualize the complete surface, you must ensure that your parameterization covers the entire complex plane, not just a portion.The complex plane 
is defined by 
, where 
is the magnitude 
and θ is the angle 
.
is defined by , where is the magnitude and θ is the angle . The square root function has two branches:
- 
- 
To visualize both branches and the symmetry, ensure you cover θ from 
to 
and r over the desired range.
to and r over the desired range.% Define the range for r and theta
r_min = 0; r_max = 4; % You can adjust the range as needed
theta_min = -pi; theta_max = pi;
% Create a grid of (r, theta)
r = linspace(r_min, r_max, 100); % 100 points in r
theta = linspace(theta_min, theta_max, 100); % 100 points in theta
[R, Theta] = meshgrid(r, theta);
% Calculate the first branch w01 and second branch w02
W01 = sqrt(R) .* exp(1i * Theta / 2); % first branch
W02 = sqrt(R) .* exp(1i * (Theta + 2*pi) / 2); % second branch
% Split into real and imaginary parts for plotting
x1 = real(W01); y1 = imag(W01); z1 = R; % First branch
x2 = real(W02); y2 = imag(W02); z2 = R; % Second branch
% Plot the first branch
figure;
surf(x1, y1, z1);
hold on;
% Plot the second branch
surf(x2, y2, z2);
% Labels and title
xlabel('Re(w)');
ylabel('Im(w)');
zlabel('r');
title('Riemann Surface of w = \surd{z}');
grid on;
hold off;
