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.
