Integrating legendre polynomials with weighting function

[fantispug]

Homework Statement

I recently came across this integral while doing a problem in electromagnetism (I'm not sure if there exists a nice analytic answer):

$$\int_{0}^{\pi}P_m(\cos(t))P_n(\cos(t)) \sin^2(t) = \int_{-1}^{1}P_m(x) P_n(x) \sqrt{1-x^2},$$

Homework Equations

P_m(x) is the m^th Legendre Polynomial.

The Attempt at a Solution

There are lots of close integrals in Gradshteyn and Ryzhik 7.1-7.2 but nothing close enough for me to use.

One way to evaluate it would be to expand the square root as a Taylor series, and then change basis to re-expand it as a series of Legendre polynomials, then use tricks involving triple integrals of Legendre polynomials (such as those in Arfken and Weber 12.9). However this is incredibly messy and I can't see how I could get an analytic expression from it.

Can anyone think of a nice way of approaching this integral? I can't use for example differentiation under the integral because I don't know how to integrate the product of Legendre functions on intervals other than [-1,1] (and [0,1]).

 

[vela]

Never mind...