% CONVERGE.M
%
% Let z be a complex number. We know z^n converges if ||z|| < 1, and that
% z^n goes to infinity if ||z|| > 1. However, if ||z|| = 1, then some
% interesting behavior occurs. We will examine this behavior graphically
% in the complex plane.
%
% Jessica Bernier        MATH215        Aug. 28, 2008

fprintf(['\nLet z be a complex number of the form z = a + bi,\n   where a and b are real numbers and i = sqrt(-1).\n']);
fprintf(['   Consider a and b in the range [-1,1] so that ||z|| = 1.\n']);

a = input('Enter the real component of the complex number (a): ');
while abs(a)>1
    a = input('Error: Please enter a value between -1 and 1: ');
end
b = sqrt(abs(1-a^2));
disp(['Given a = ', num2str(a), ', we need b = +/-', num2str(b), '.']);
disp(['The ratio (b^2)/(a^2) = ', num2str((b/a)^2), '.']);

nmax = 500; %This is the maximum value of n to use. Adjust nmax to show more or fewer points, if needed.
z = zeros(nmax,1);
z(1) = complex(a,b);
phi = angle(z(1));

figure;
title('Behavior of z^n on the complex plane, for ||z|| = 1')
xlabel('Real component (a)')
ylabel('Imaginary component (b)')
hold on; axis equal; grid on;
plot(complex(cos(0:pi/1000:2*pi),sin(0:pi/1000:2*pi)),'r'); %Plot the unit circle in red, as a guide.

for n=1:nmax
    z(n) = exp(i*phi*n); %The radius will always be 1.
    if isreal(z(n))==1 %If Matlab considers z to be real, it will graph in Cartesian coordinates. We need to manually declare z to be complex. This occurs for a = 1.
        plot(complex(z(n)), '.'); %complex(z(n)) = z(n) + 0i for real z(n).
    else
        plot(z(n), '.');
    end
    pause(10/nmax) %Delay updating the figure so we can observe each new point being plotted. It takes ~10 seconds to complete the figure.
end
hold off

disp('All done!');