3-j symbol ($\left(\begin{array}{ccc}{j}_1& j_2& j_3\\ m_1& m_2& m_3\end{array}\right)$)

A symmetrized form of the Clebsch-Gordan coefficients used for coupling three angular momenta. The relation is $⟨j_1{m}_1,j_2{m}_2|j_3{m}_3⟩=(-1{)}^{j_1-j_2+m_3}\sqrt{2j_3+1}\left(\begin{array}{ccc}{j}_1& j_2& j_3\\ m_1& m_2& -m_3\end{array}\right)$. The symbol vanishes unless the magnetic indices sum to zero and the angular momenta obey the triangle and integrality conditions. Products of two such symbols express the Gaunt integral; the zero-magnetic-index symbol enforces even total multipole for integer spherical harmonics.