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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.

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  • $\begingroup$ Please add the exact code you've tried $\endgroup$ – MarcoB Mar 11 at 4:01
  • $\begingroup$ Have you seen this sign convention issue in Mathematica? $\endgroup$ – Roman Mar 11 at 5:47
  • $\begingroup$ Added the Mathematica code for $\tau$ that I've used. $\endgroup$ – hingramo Mar 12 at 3:02
  • $\begingroup$ Are you sure the factor $n(n+1)$ only multiplies one of the Wigner D functions and not the other? The formula looks wrong, maybe there's a typo in the book. $\endgroup$ – Roman Mar 12 at 13:37
  • $\begingroup$ That's the way it's defined in referenced the textbook but you're right that it seems odd. I will look for another source. $\endgroup$ – hingramo Mar 12 at 15:32

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