% Matlab script to plot associated functions and polynomials
%
clear all;
t = [-1:0.05:1];
P2 = legendre(2,t);
P3 = legendre(10,t);
P4 = legendre(40,t);
P5 = legendre(50,t);
%

% Now do sectoral Harmonics (m=n)
figure(1)
plot(t,P2(3,:),'b');
hold on;
plot(t,P3(11,:),'g');
plot(t,P4(41,:),'r');
plot(t,P5(51,:),'m');
title('Sectoral harmonics: Degrees 2-50, 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,'S2 b, S10 g, S40 r, S50 m')
%
% Now "normalize" the sectorals
norm = zeros(5,1);
i=2;  norm(2) = sqrt(2*gamma(2*i+1)/(2*i+1));
i=10; norm(3) = sqrt(2*gamma(2*i+1)/(2*i+1));
i=40; norm(4) = sqrt(2*gamma(2*i+1)/(2*i+1));
i=50; norm(5) = sqrt(2*gamma(2*i+1)/(2*i+1));


figure(2)
plot(t,P2(3,:)/norm(2),'b');
hold on;
plot(t,P3(11,:)/norm(3),'g');
plot(t,P4(41,:)/norm(4),'r');
plot(t,P5(51,:)/norm(5),'m');
title('Normalized Sectoral harmonics: Degrees 2-50, 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,'S2 b, S10 g, S40 r, S50 m')