ABSTRACT
Pn(cos v) dα, (14.3.27)
where cos v=cosλ cosμ+ sinλ sinμ cosα, the product can be rewritten in the form
f˜(n) g˜(n) = 1
π
π∫
f(cosμ) sinμ
× ⎡ ⎣
π∫
π∫
g(cosμ)Pn(cosμ) sinλ dα dλ
⎤ ⎦dμ. (14.3.28)
We next use Churchill and Dolph’s (1954, pp. 94-96) geometrical arguments to replace the double integral inside the square bracket by
π∫
π∫
g(cosμ cos v + sinμ sin v cosβ)Pn(cos v) sin v dv. (14.3.29)
Substituting this result in (14.3.26) and changing the order of integration, we obtain
f˜(n) g˜(n) = 1
π
π∫
Pn(cos v) sin v
⎡ ⎣
π∫
π∫
f(cosμ) sinμ g(cosλ)dμ dβ
⎤ ⎦dv
=
π∫
h(cos v)Pn(cos v) sin v dv, (14.3.30)
cosλ=cosμ cos v + sinμ sin v cosβ, (14.3.31)
and
h(cos v) = 1
π
π∫
f(cosμ) sinμ dμ
π∫
g(cosλ) dβ.
