体球調和関数
次に、x = r sinθcosφ と y = r sinθsinφ に注意すると
r m sin m  θ cos  m φ = 1 2 [ ( r sin  θ e i φ ) m + ( r sin  θ e − i φ ) m ] = 1 2 [ ( x + i y ) m + ( x − i y ) m ] {\displaystyle r^{m}\sin ^{m}\theta \cos m\varphi ={\frac {1}{2}}\left[(r\sin \theta e^{i\varphi })^{m}+(r\sin \theta e^{-i\varphi })^{m}\right]={\frac {1}{2}}\left[(x+iy)^{m}+(x-iy)^{m}\right]}
同様に
r m sin m  θ sin  m φ = 1 2 i [ ( r sin  θ e i φ ) m − ( r sin  θ e − i φ ) m ] = 1 2 i [ ( x + i y ) m − ( x − i y ) m ] . {\displaystyle r^{m}\sin ^{m}\theta \sin m\varphi ={\frac {1}{2i}}\left[(r\sin \theta e^{i\varphi })^{m}-(r\sin \theta e^{-i\varphi })^{m}\right]={\frac {1}{2i}}\left[(x+iy)^{m}-(x-iy)^{m}\right].}
l = 5 以下の関数の明示式を記す。ここで
Π ̄ l m ( z ) ≡ [ ( 2 − δ m 0 ) ( l − m ) ! ( l + m ) ! ] 1 / 2 Π l m ( z ) . {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)\equiv \left[{\tfrac {(2-\delta _{m0})(\ell -m)!}{(\ell +m)!}}\right]^{1/2}\Pi _{\ell }^{m}(z).}