MATLAB: Legendre polynomial Symbolic derivative

Question

I need to find derivative with respect to the argument of Associated Legendre functions symbolically . The following does not work:

x=sym('x', 'real');

q=sym('q', 'positive'); % harmonic index

t=sym('t', 'real');

 x=cos(t);
 P=legendre(q,x )% Associated Legendre function  
 P_1st_der=diff(P,x)

Can anyone help me?

Thanks

Irfan

Answer 1

That's a bit messy because the MUPAD legendre() function returns an array of all the associated Legendre values of order 0 to q.

When LegendreP(q, u, f(x)) is the associated Legendre function of degree q and order u, u from 0:q, then

diff(LegendreP(q, u, f(x)), x)

is

((q-u+1)*LegendreP(q+1, u, f(x))-(q+1)*f(x)*LegendreP(q, u, f(x)))*(diff(f(x), x))/(f(x)^2-1)

If x is a vector then that would probably have to be

((q-u+1)*LegendreP(q+1, u, f(x))-(q+1)*f(x).*LegendreP(q, u, f(x))).*(diff(f(x), x)) ./ (f(x).^2-1)

To construct an entire matrix of these symbolically over u = 0:q, you would have to throw in some repmat() on the portions having to do with f(x) appearing outside of your corresponding legendre() calls (since legendre() returns a matrix so you would have to copy f(x) q+1 times in order to do the element-wise multiplication and division)

It would be easier if you only needed one particular degree and order -- e.g.,

LegendreP(q+1, u, f(x))

would become

 Lq1 = legendre(q+1,f(x));

then use

 Lq1(u+1,:)