# 3.37. Wigner D Function¶

The Wigner function gives the matrix elements of the rotation operator in the -representation. For the Euler angles , , , the function is defined as:

Where the rotation operator is defined using the -- convention:

Here is the projection of the total angular momentum on an -axis. The is the eigenstate of the operators and . Using the fact that , we can see that the Wigner function can always be written using the Wigner small- function as:

where

We can use the following relations to evaluate :

## 3.37.1. Derivation¶

The small- function formula above can be derived from the following formula:

by substituting

into

This follows from:

let the polynomial be:

and (using binomial theorem in the process):

And it is the coefficient of .