Question

\[ \int_0^\pi P_n( \cos \theta ) \cos (n \theta ) d\theta \]

Can anyone tell me how I can get started with evaluating this integral?

Answer 1

I think you need to use \(\large{\cos(x)\cos(y)=\frac{1}{2}(\cos(x+y)+\cos(x-y)).}\)

Answer 2

How would that help in dealing with the Legendre polynomial?

Answer 3

Oh you should have said that \(P_n\) represents the Legendre polynomial!

Answer 4

Let \(\large{I_n=\int_0^{\pi} P_n\cos{x}\cos(nx)dx}\), then:
\[I_0=\int\limits_0^\pi P_0\cos {x} \cos (0)dx=\int\limits_0^\pi \cos{x}dx=\sin{x}|_0^{\pi}=0.\]
\(\large I_1=\int\limits_0^{\pi}P_1\cos^2 x dx=\int\limits_{0}^{\pi}x \cos^2{x}dx\). Using integration by parts \(\large I_1=\frac{\large \pi^2}{4}\).

Answer 5

Similarly you can find \(\large I_2, I_3,..\). Some of the integrals will be lengthy.