```function PPDemo(cs)
% demonstration of phase plots with various color schemes
%
% Usage: PPDemo, PPDemo(cs)
%
% cs - color scheme (for example 'p' or 'pn')
%
% call PPDemo without argument to see a demonstration of available color schemes
% call 'help colscheme' to get a list of available color schemes

% Part of the Complex Function Explorer (former Phase Plot) package
% Version 1.1, February 1, 2014
% Copyright(c) 2012-2014 by Elias Wegert (elias@wegert.com, www.wegert.com)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% if there are memory problems reduce the resolution

xres = 1000;
yres = 1000;

%%

z = zdomain(-2-2i,2+2i,xres,yres);
w = (z-1)./(z.*z+z+1);

if nargin == 1;
%if strcmp(cs,'f') || strcmp(cs,'fn'), w=z+1./z; end
PhasePlot(z,w,cs);
else

disp(' ')
disp('VISUAL EXPLORATION OF COMPLEX FUNCTIONS USING PHASE PLOTS')
disp(' ')
disp('Phase plots depict the color-coded phase (argument) of complex')
disp('functions as images on their domains.')
disp('Several modifications allow one to read off properties more easily.')
disp(' ')
disp('For detailed information please consult the textbook')
disp('E. Wegert: Visual Complex Functions, Springer/Birkhaeuser 2012')
pause

%% proper phase plot

figure

PhasePlot(z,w,'p');
title('p - proper phase plot, hsv coloring')

disp(' ')
disp('To create a phase plot of w=f(z), call PhasePlot(z,w,cs), where')
disp('z  is a 2D field of complex numbers covering the domain of f')
disp('w  is a 2D field of corresponding values f(z)')
disp('cs is a colorscheme (optional), for example cs=''p''')
disp('Call ''help colscheme'' to get a list of available color schemes.')
disp('Call PPDemo(cs) to see the standard example represented by color scheme cs.');
pause

disp(' ')
disp('The function f(z)=(z-1)/(z^2+z+1) is depicted in |Re(z)|<2, |Im(z)|<2')
pause

PhasePlot(z,w,'pn');
title('pn - proper phase plot, hsv coloring modified according to NIST standard')
disp(' ')
disp('Note that all color schemes can be modified according to the')
disp('NIST standard by appending ''n'' to cs, for example cs=''pn''');
pause

%% colored modifications

PhasePlot(z,w,'m');
title('m - enhanced phase plot with modulus jumps')
disp(' ')
disp('There are various modifications of the standard color scheme.')
disp('Here we use shading to generate contour lines of |f(z)|.')
pause

PhasePlot(z,w,'c');
title('c - phase plot with conformal polar grid')
disp(' ')
disp('Here we further added contour lines of arg f(z)')
disp('which generates tiles of almost square shape.')
pause

PhasePlot(z,w,'j');
title('j - phase plot with some enhanced isochromatic lines')
disp(' ')
disp('The contour lines of arg f(z) are isochromatic.')
pause

%PhasePlot(z,w,'q');
%title('q - phase plot colored in steps ')

PhasePlot(z,w,'d');
title('d - standard domain coloring')
disp(' ')
disp('This is standard domain coloring, where color represents the phase')
disp('and shading corresponds to the modulus of f(z).')
disp('In dark regions |f(z)| is small, in bright regions |f(z)| is large.')
pause

PhasePlot(z,w,'e');
title('e - enhanced domain coloring')
disp(' ')
disp('Here we see enhanced domain coloring with contour lines of |f(z)|.')
pause

%% black and white modifications

PhasePlot(z,w,'u');
title('u - polar chessboard')
disp(' ')
disp('The color scheme ''u'' is good to visualize conformal mappings.')
pause

PhasePlot(z,w,'a');
title('a - alternating black and white phase')
disp(' ')
disp('The color scheme ''a'' is appropriate to depict electric field lines.')
pause
hold on
plot3(cos(2*pi/3),sin(2*pi/3),20,'r.','Markersize',20);
plot3(cos(4*pi/3),sin(4*pi/3),20,'r.','Markersize',20);
plot3(1,0,1,'b.','Markersize',20);
disp('Here we have one negative and two positive point charges.')
hold off
pause

PhasePlot(z,w,'b');
hold on
plot3(cos(2*pi/3),sin(2*pi/3),20,'r.','Markersize',20);
plot3(cos(4*pi/3),sin(4*pi/3),20,'r.','Markersize',20);
plot3(1,0,1,'b.','Markersize',20);
title('b - alternating black and white modulus')
disp(' ')
disp('Using the scheme ''b'' highlights the corresponding potential lines.')
hold off
pause

%% color schemes based on cartesian coordinates

PhasePlot(z,w,'v');
title('v - Cartesian chessboard')

%% color scheme for depicting stream lines of potential flows

disp(' ')
disp('The scheme ''f'' is designed for depicting stream lines of potential flows.')
disp('Here f(z)=(z+1/z)/2 is the Joukovski function.')
hold on
PhasePlot(z,z+1./z,'f');
pause
t = exp(1i*linspace(0,2*pi,1000));
patch(real(t),imag(t),20*ones(size(t)),'k');
disp('The function in the interior of the unit circle is irrelevant.')
title('Plane potential flow around a circular cylinder')
hold off
pause

clf
disp(' ')
disp('Phase plots can also be depicted on the Riemann sphere using the routine PPOnSphere.');
disp('This code uses one chart to (almost) cover the sphere:')
disp(' ')
disp('z = zplanePP; w = (z-1)./(z.^2+z+1); PPOnSphere(z,w);')

z = zplanePP;
w = (z-1)./(z.^2+z+1);
PPOnSphere(z,w);
pause

disp(' ')
disp('Using two charts to cover the sphere yields a more uniform distribution of points:');
disp(' ')
disp('z1= zdomain(-1-1i,1+1i); w1=(z1-1)./(z1.^2+z1+1); PPOnSphere(z1,w1);')
clf, hold on
z1 = zdomain(-1-1i,1+1i,1000,1000);
w1=(z1-1)./(z1.^2+z1+1);
PPOnSphere(z1,w1);
axis(0.8*[-1,1,-1,1,-1,1])
pause

disp('z2=1./z1; w2=(z2-1)./(z2.^2+z2+1); hold on, PPOnSphere(z2,w2);')
z2=1./z1;
w2=(z2-1)./(z2.^2+z2+1);
PPOnSphere(z2,w2);
axis(0.8*[-1,1,-1,1,-1,1])
hold off
pause

disp(' ')
disp('PPOnSphLow(z1,w1) shows the lower hemisphere projected to the plane.');
PPOnSphLow(z1,w1);
pause

disp('PPOnSphUp(z2,w2)  shows the upper hemisphere projected to the plane.');
PPOnSphUp(z2,w2);
pause

disp(' ')
disp('Phase plots can also be depicted on analytic landscapes.')
disp('The graphical user interface CFEGUI allows one to control parameters.')
pause

disp(' ')
disp('Now invoking the interface CFEGUI for exploring complex functions.')
disp('Make yourself familiar with some examples using the Load button.')

CFEGUI

end

```