How do I properly implement angular functions using Wigner D-functions in Mathematica? Angular functions are commonly used in light scattering calculations and can be defined using Wigner D-functions. The definitions in terms of Legendre polynomials $P^m_n$ are $$ \tau_{mn1} (\cos \vartheta ) = \frac{d}{d \vartheta} P^m_n (\cos \vartheta ), \quad \tau_{mn2} (\cos \vartheta ) = \frac{m}{\sin \vartheta} P^m_n (\cos \vartheta ), $$ but they can also be expressed using the Wigner D-functions $\mathcal{D}^s_{mn}$ with $$ \tau_{mnp} = - \frac{1}{2} [ n(n+1) \mathcal{D}_{1n}^m + (-1)^p \mathcal{D}_{-1n}^m]. $$ However, I cannot get the results from this last equation to match the results from the Legendre polynomial definition using the built in functions in Mathematica.

Note that the above definitions are from pages 230 and 233 of the book "Light Scattering by Nonspherical Particles" by M. Mishchenko et al. 2000.

How do I properly implement angular functions using Wigner D-functions in Mathematica? Angular functions are commonly used in light scattering calculations and can be defined using Wigner D-functions. The definitions in terms of Legendre polynomials $P^m_n$ are $$ \tau_{mn1} (\cos \vartheta ) = \frac{d}{d \vartheta} P^m_n (\cos \vartheta ), \quad \tau_{mn2} (\cos \vartheta ) = \frac{m}{\sin \vartheta} P^m_n (\cos \vartheta ), $$ but they can also be expressed using the Wigner D-functions $\mathcal{D}^s_{mn}$ with $$ \tau_{mnp} = - \frac{1}{2} [ n(n+1) \mathcal{D}_{1n}^m + (-1)^p \mathcal{D}_{-1n}^m]. $$ However, I cannot get the results from this last equation to match the results from the Legendre polynomial definition using the built in functions in Mathematica.

The way I've defined the $\tau$ function in Mathematica is:

\[Tau][n_, m_, p_, \[Theta]_] := -(1/ 2)*(n*(n + 1)*WignerD[{n, 1, m}, \[Theta]] + (-1)^p* WignerD[{n, -1, m}, \[Theta]]) 

Note that the above definitions are from pages 230 and 233 of the book "Light Scattering by Nonspherical Particles" by M. Mishchenko et al. 2000.