gives the values of the Wigner 3 symbol.


  • The 3 symbols vanish except when and the satisfy a triangle inequality.
  • The parameters of ThreeJSymbol can be integers, halfintegers or symbolic expressions.
  • The 3 symbols and ClebschGordan coefficients in the Wolfram Language satisfy the relation .


open allclose all

Basic Examples  (2)

Evaluate a 3 symbol with integer arguments:

Use symbolic arguments to obtain exact symbolic answers:

Scope  (2)

ThreeJSymbol works with integer and halfinteger arguments:

TraditionalForm formatting:

Applications  (3)

Plot 3 symbol coefficients as a function of and :

Apply angular momentum operators to spherical harmonics:

Evaluate Gaunt coefficients:

Evaluate as an explicit triple integral of spherical harmonics:

Properties & Relations  (4)

Evaluate the completely symbolic case of ThreeJSymbol:

Demonstrate sum orthogonality:

Convert a 3 symbol to a Regge R symbol:

Check the symmetry of the Regge R symbol:

Compare with the sum definition:

Possible Issues  (1)

A message is issued and the result 0 returned in the case :

Wolfram Research (1991), ThreeJSymbol, Wolfram Language function,


Wolfram Research (1991), ThreeJSymbol, Wolfram Language function,


Wolfram Language. 1991. "ThreeJSymbol." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (1991). ThreeJSymbol. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_threejsymbol, author="Wolfram Research", title="{ThreeJSymbol}", year="1991", howpublished="\url{}", note=[Accessed: 29-May-2024 ]}


@online{reference.wolfram_2024_threejsymbol, organization={Wolfram Research}, title={ThreeJSymbol}, year={1991}, url={}, note=[Accessed: 29-May-2024 ]}