Substitute the P_l(x), you found in Problems 4.3 or 5.3 into equation (10.6)to find P_l^mthen let x=cosθto evaluate:

P₄¹(cosθ)

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)

Putting l=4 and m=1:

P₄¹(x)=(1-x²)^1/2 d¹/dx¹ P₄(x)

P₄¹(x)=(1-x²)^1/2 d/dx P₄(x)

P₄¹(x)=(1-x²)^1/2 d/dx (35x⁴/8-15x²/4+3/8)

P₄¹(x)= 1/2 (1-x²)^1/2 d/dx (35x³-15x) ..... (2)

On substituting x=cosx into equation (2):

P₄¹ (x)=1/2 (1-x²)^1/2 (35x³-15x)

P₄¹ (cosθ)=1/2 (1-cos²θ)^1/2 (35 cos³-15 cosθ)

P₄¹ (cosθ)=1/2 (sin²θ)^1/2 (35 cos³-15 cosθ) P

P₄¹ (cosθ)=1/2 (sinθ) (35 cos³-15 cosθ)

Therefore, the value of P₄¹ (cosθ) is found to be 1/2 (sinθ) (35 cos³θ-15 cosθ).