content_image/337164.jpg content_image/337162.jpg content_image/337163.jpg content_image/337161.jpg content_image/337160.jpg content_image/337159.jpg content_image/337158.jpg content_image/337157.jpg
Hence 

<img src='content_image/337164.jpg'>

where the sum involving associated Legendre functions of the first kind vanish in the inte- gration over φ ′ . Now we introduce the notation 

So 

<img src='content_image/337162.jpg'>

<img src='content_image/337163.jpg'>

with f = P$_{2}$ (cos θ ′ ). From the orhogonality conditions of Legendre polynomials [79] we have that 〈 P$_{2}$$_{n}$ 〉 f = 2 / (2 m + 1) δ$_{mn}$ where δ$_{mn}$ is the Kronecker- δ symbol. Here, m = 2 and, given that c$_{2}$ = − 5 π/ 32, we have 

<img src='content_image/337161.jpg'>

Hence 

<img src='content_image/337160.jpg'>

as required. The last integral to consider is 

<img src='content_image/337159.jpg'>

We consider integrating the first term of (A18), namely 

<img src='content_image/337158.jpg'>

We let ω ′′ = e$_{z}$ in the integrand so that 

<img src='content_image/337157.jpg'>

