Question

Substitute the P₁(x)you found in Problems 4.3 or 5.3 into equation (10.6)to find, P_l^m(x); then let x=cos θto evaluate:

P₃²(cosθ)

Short answer

The value of P₃²(cosθ) is 15 cosθsin²θ.

Step-by-step solution

Concept:

An equation similar to Legendre’s equation is-

(1-x²) y"-2xy'+[l(l+1)-m²/1-x²] y=0 (∀ m²≤ l²)

The solution to this equation is of the form

P_l^m (x)=(1-x²)^m/2 d^m/dx^m (P_l(x))

These solutions are called Associated Legendre Functions.

It is given that,

P_l^m(x) =(1-x²)^m/2 d^m/dx^m P_l(x) …… (1)

Put l=3 and m=2 in equation (1):

Putting l=3 and m=2:

P₃²(x) = (1-x²)^2/2 d²/dx² P₃(x)

P₃²(x) = (1-x²)¹ d²/dx² (1/2 (5x²-3x))

P₃²(x) = (1-x²) (15x)

Substitute cosθ for x as,

P₃²(cosθ) = (1-cos²θ) 15 cosθ

=15 cosθ sin²θ