% Matlab script to plot associated functions and polynomials
%
clear all;
t = [-1:0.05:1];
P2 = legendre(2,t);
P3 = legendre(3,t);
P4 = legendre(4,t);
P5 = legendre(5,t);
%
% Now make plots
clf
figure(1)
plot(t,P2(1,:),'b');
hold on;
plot(t,P3(1,:),'g');
plot(t,P4(1,:),'r');
plot(t,P5(1,:),'m');
title('Polynomials: Degrees 2-5; Order 0')
xlabel('Cos(theta)'); ylabel('Legendre Function')
yt = get(gca,'YLIM');
ypos = yt(1) + (yt(2)-yt(1))*0.05;
text(-0.95,ypos,'P2 b, P3 g, P4 r, P5 m')
%
% Now do m=1
figure(2)
plot(t,P2(2,:),'b');
hold on;
plot(t,P3(2,:),'g');
plot(t,P4(3,:),'r');
plot(t,P5(4,:),'m');
title('Asociated Function: Degrees 2-5, order 1')
xlabel('Cos(theta)'); ylabel('Legendre Function')
yt = get(gca,'YLIM');
ypos = yt(1) + (yt(2)-yt(1))*0.05;
text(-0.95,ypos,'P2 b, P3 g, P4 r, P5 m')
% Now do m=2
figure(3)
plot(t,P2(3,:),'b');
hold on;
plot(t,P3(3,:),'g');
plot(t,P4(3,:),'r');
plot(t,P5(3,:),'m');
title('Asociated Function: Degrees 2-5, order 2')
xlabel('Cos(theta)'); ylabel('Legendre Function')
yt = get(gca,'YLIM');
ypos = yt(1) + (yt(2)-yt(1))*0.05;
text(-0.95,ypos,'P2 b, P3 g, P4 r, P5 m')
% Now do m=3
figure(4)
plot(t,P3(3,:),'g');hold on;
plot(t,P4(3,:),'r');
plot(t,P5(3,:),'m');
title('Asociated Function: Degrees 2-5, order 3')
xlabel('Cos(theta)'); ylabel('Legendre Function')
yt = get(gca,'YLIM');
ypos = yt(1) + (yt(2)-yt(1))*0.05;
text(-0.95,ypos,'P3 g, P4 r, P5 m')
% Now do m=4
figure(5)
plot(t,P4(3,:),'r');hold on;
plot(t,P5(3,:),'m');
title('Asociated Functions: Degrees 2-5, order 4')
xlabel('Cos(theta)'); ylabel('Legendre Function')
yt = get(gca,'YLIM');
ypos = yt(1) + (yt(2)-yt(1))*0.05;
text(-0.95,ypos,'P4 r, P5 m')
% Now do sectoral Harmonics (m=n)
figure(6)
plot(t,P2(3,:),'b');
hold on;
plot(t,P3(4,:),'g');
plot(t,P4(5,:),'r');
plot(t,P5(6,:),'m');
title('Sectoral harmonics: Degrees 2-5, order m=n')
xlabel('Cos(theta)'); ylabel('Legendre Function')
yt = get(gca,'YLIM');
ypos = yt(1) + (yt(2)-yt(1))*0.05;
text(-0.95,ypos,'P2 b, P3 g, P4 r, P5 m')
%
% Now "normalize" the sectorals
for i = 2:5
	norm(i) = sqrt(2*gamma(2*i+1)/(2*i+1));
end
figure(7)
plot(t,P2(3,:)/norm(2),'b');
hold on;
plot(t,P3(4,:)/norm(3),'g');
plot(t,P4(5,:)/norm(4),'r');
plot(t,P5(6,:)/norm(5),'m');
title('Normalized Sectoral harmonics: Degrees 2-5, order m=n')
xlabel('Cos(theta)'); ylabel('Legendre Function')
yt = get(gca,'YLIM');
ypos = yt(1) + (yt(2)-yt(1))*0.05;
text(-0.95,ypos,'P2 b, P3 g, P4 r, P5 m')