It shows in the image to take the real part of the functions, but your description says something different.
The code below is according to the image as reference -
"Please tell me how to draw these in MATLAB (with better colors)"
You figured out using fsurf, for colors, experiment with colormap, choose the one you like the most.
syms z r theta
X(z) = real(-i/2*log((1+z*i)/(1-z*i)));
Y(z) = real(-i/2*log((1+z)/(1-z)));
Z(z) = real(1/2*log((1+z^2)/(1-z^2)));
X(r,theta)=subs(X,z,r*exp(1i*theta));
Y(r,theta)=subs(Y,z,r*exp(1i*theta));
Z(r,theta)=subs(Z,z,r*exp(1i*theta));
fsurf(X,Y,Z,[0 1 pi/24 2*pi+pi/24],'MeshDensity',150,'LineStyle','none')
colormap(cool)
%Adjust x, y and z axis limits according to the reference image
axis([-1 1 -1 1 -1 1])
"Also, i wanted to understand the significance of the above theta min and theta max setup."
The values provided are used to calculate the values of X, Y and Z functions and then the values are used to plot the function. The plot will be according to the values provided, if no values are provided, default values are used.
"How will we know what to set pi/24 or pi/32?"
You can choose anyvalue, as using [value 2*pi+value] is equivalent to using [0 2*pi]. (I don't know the reason why the image you got for [0 2*pi] is so different from [pi/24 2*pi+pi/24])
See the below plot and compare it to the above plot -
figure
fsurf(X,Y,Z,[0 1 0 2*pi],'MeshDensity',150,'LineStyle','none')
colormap(cool)
axis([-1 1 -1 1 -1 1])
You can also compare the data, which you can get from the handle of fsurf().
