DLMF:34.2.E4 (Q9713): Difference between revisions

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Property / Symbols defined: $$\mathit{3j}$$ symbol / rank
 
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Revision as of 15:29, 30 December 2019

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DLMF:34.2.E4
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    3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 = ( - 1 ) j 1 - j 2 - m 3 ⁢ Δ ⁢ ( j 1 ⁢ j 2 ⁢ j 3 ) ⁢ ( ( j 1 + m 1 ) ! ⁢ ( j 1 - m 1 ) ! ⁢ ( j 2 + m 2 ) ! ⁢ ( j 2 - m 2 ) ! ⁢ ( j 3 + m 3 ) ! ⁢ ( j 3 - m 3 ) ! ) 1 2 ⁢ ∑ s ( - 1 ) s s ! ⁢ ( j 1 + j 2 - j 3 - s ) ! ⁢ ( j 1 - m 1 - s ) ! ⁢ ( j 2 + m 2 - s ) ! ⁢ ( j 3 - j 2 + m 1 + s ) ! ⁢ ( j 3 - j 1 - m 2 + s ) ! , threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 superscript 1 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑚 3 Δ subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 superscript subscript 𝑗 1 subscript 𝑚 1 subscript 𝑗 1 subscript 𝑚 1 subscript 𝑗 2 subscript 𝑚 2 subscript 𝑗 2 subscript 𝑚 2 subscript 𝑗 3 subscript 𝑚 3 subscript 𝑗 3 subscript 𝑚 3 1 2 subscript 𝑠 superscript 1 𝑠 𝑠 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 𝑠 subscript 𝑗 1 subscript 𝑚 1 𝑠 subscript 𝑗 2 subscript 𝑚 2 𝑠 subscript 𝑗 3 subscript 𝑗 2 subscript 𝑚 1 𝑠 subscript 𝑗 3 subscript 𝑗 1 subscript 𝑚 2 𝑠 {\displaystyle{\displaystyle\mathit{3j}{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{% 3}}={(-1)^{j_{1}-j_{2}-m_{3}}}\Delta(j_{1}j_{2}j_{3})\left((j_{1}+m_{1})!(j_{1% }-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!(j_{3}+m_{3})!(j_{3}-m_{3})!\right)^{% \frac{1}{2}}\*\sum_{s}\frac{(-1)^{s}}{s!(j_{1}+j_{2}-j_{3}-s)!(j_{1}-m_{1}-s)!% (j_{2}+m_{2}-s)!(j_{3}-j_{2}+m_{1}+s)!(j_{3}-j_{1}-m_{2}+s)!},}}
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    DLMF:34.2.E4
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