スレーター積分
∑ m c k ( l , m + r , l ′ , m ) c k ( l , m + r , l ~ , m ) = δ l , l ~ ⋅ ( 2 l + 1 ) ( 2 l ′ + 1 ) 2 k + 1 ⋅ c k ( l , 0 , l ′ , 0 ) . ∑ m c k ( l , m + r , l ′ , m ) c q ( l , m + r , l ′ , m ) = δ k , q ⋅ ( 2 l + 1 ) ( 2 l ′ + 1 ) 2 k + 1 ⋅ c k ( l , 0 , l ′ , 0 ) .
この積分は次のような[恒等式]をもつ
c k ( l , m , l ′ , m ′ ) = c k ( l , − m , l ′ , − m ′ ) = ( − 1 ) m − m ′ c k ( l ′ , m ′ , l , m ) = ( − 1 ) m − m ′ 2 l + 1 2 k + 1 c l ( l ′ , m ′ , k , m ′ − m ) = ( − 1 ) m ′ 2 l ′ + 1 2 k + 1 c l ′ ( k , m − m ′ , l , m ) . ∑ m = − l l c k ( l , m , l , m ) = ( 2 l + 1 ) δ k , 0 . ∑ m = − l l ∑ m ′ = − l ′ l ′ c k ( l , m , l ′ , m ′ ) 2 = ( 2 l + 1 ) ( 2 l ′ + 1 ) ⋅ c k ( l , 0 , l ′ , 0 ) . ∑ m = − l l c k ( l , m , l ′ , m ′ ) 2 = 2 l + 1 2 l ′ + 1 ⋅ c k ( l , 0 , l ′ , 0 ) . ∑ m = − l l c k ( l , m , l ′ , m ′ ) c k ( l , m , l ~ , m ′ ) = δ l ′ , l ~ ⋅ 2 l + 1 2 l ′ + 1 ⋅ c k ( l , 0 , l ′ , 0 ) .