Orthogonality relationship 中)Ynm(,φ) sin e de de=8 s,8, (E182) ∑∑Ym(,中)Ym(0,)=8(-9)6(c0s6-cos (E.183) Y∞(6,) (E184) Y10(6,中) (E185) Y116,中) (E.186) Yx(,小)=V+(co0-1 (E187) ￥21(,中) sin e cos be中 (E188) Y22(,y y32 sin2Be2js Y30(,中)= 31(6,中) sin 0(5 cos 26-1)e Y(6,中)=V32x e cos Be2jo (E192) Y3(,中)=-V64 sin pejo (E193) Functional relationships v 43 Yn,-m(6,中)=(-1)Ymn(6,中) (E195) Addition formulas Pn(cosy)= >Ym(.d)Y (E.196) Pn(cos y)= Pn(cos 8)Pn(cos 8)+ =(n+m)/pm(cos 0)Pm" (cos 0)cos[m(o-pI (E.197) @2001 by CRC Press LLCOrthogonality relationships  π −π  π 0 Y ∗ n m(θ, φ)Ynm(θ, φ)sin θ dθ dφ = δn nδm m (E.182) ∞ n=0 n m=−n Y ∗ nm(θ , φ )Ynm(θ, φ) = δ(φ − φ )δ(cos θ − cos θ ) (E.183) Specific examples Y00(θ, φ) =  1 4π (E.184) Y10(θ, φ) =  3 4π cos θ (E.185) Y11(θ, φ) = − 3 8π sin θe jφ (E.186) Y20(θ, φ) =  5 4π 3 2 cos2 θ − 1 2 (E.187) Y21(θ, φ) = − 15 8π sin θ cos θe jφ (E.188) Y22(θ, φ) =  15 32π sin2 θe2 jφ (E.189) Y30(θ, φ) =  7 4π 5 2 cos3 θ − 3 2 cos θ (E.190) Y31(θ, φ) = − 21 64π sin θ  5 cos2 θ − 1  e jφ (E.191) Y32(θ, φ) =  105 32π sin2 θ cos θe2 jφ (E.192) Y33(θ, φ) = − 35 64π sin3 θe3 jφ (E.193) Functional relationships Yn0(θ, φ) = 2n + 1 4π Pn(cos θ) (E.194) Yn,−m(θ, φ) = (−1) mY ∗ nm(θ, φ) (E.195) Addition formulas Pn(cos γ) = 4π 2n + 1 n m=−n Ynm(θ, φ)Y ∗ nm(θ , φ ) (E.196) Pn(cos γ) = Pn(cos θ)Pn(cos θ ) + + n m=−n (n − m)! (n + m)! Pm n (cos θ)Pm n (cos θ ) cos m(φ − φ )  (E.197)