The Wigner 
-functions
are the matrix
elements of the 
-dimensional
irreducible unitary group representations
of the special unitary group 
parametrized by three Euler
angles.
If 
are integers or half-integers subject to conditions 
, 
, 
, and 
are integers, and 
and 
, then the function can be evaluated to a closed-form
expression.
With the phase convention used by the Wolfram Language, the dependence on the outer Euler angles factors as
 |
(1)
|
where 
is the Wigner small-
function. Other common conventions differ by signs in the exponents and by phase
factors (Varshalovich et al. 1988).
In the Wolfram Language, WignerD[
j, m, n
, psi, theta, phi] gives 
, while WignerD[
j, m, n
, theta, phi] gives 
, which is a generalized spherical
harmonic.
The small-
function can be expressed using a Jacobi polynomial.
In particular, let 
,
,
and 
.
Then
 |
(2)
|
where the phase factor in the Wolfram Language convention is
 |
(3)
|
With normalized Haar measure 
, their orthogonality relation is
 |
(4)
|
Together with the Peter-Weyl theorem, this implies that the Wigner 
-functions form a complete
orthogonal system in the L2-space
of square-integrable functions on 
.
