I want to do some calculations which involve spin-weighted spherical harmonics of spin weight -2. How can I write that in Mathematica? Is there any built-in symbol for that?

  • 2
    $\begingroup$ Have you seen: demonstrations.wolfram.com/SpinWeightedSphericalHarmonics $\endgroup$ Commented Feb 23, 2022 at 10:26
  • $\begingroup$ Yes. I have seen it. But it doesn't show how to write it in Mathematica. I do not want to plot anything. I want to do analytical calculations, integrations that involve spin-weighted spherical harmonics of spin weight -2. $\endgroup$
    – apk
    Commented Feb 23, 2022 at 11:27
  • 1
    $\begingroup$ @apk Then please include what you've tried so far and what calculations you would like to carry out. $\endgroup$
    – MarcoB
    Commented Feb 23, 2022 at 11:34

1 Answer 1


You can download the source of the demonstrations project Daniel mentioned his comment, which contains the definition you are looking for

Y[s_, l_, m_, th_, ph_] := (-1)^m*Simplify[
  Sqrt[((l + m)!*(l - m)!*(2*l + 1))/((l + s)!*(l - s)!*4*Pi)]*
     Binomial[l - s, r]*Binomial[l + s, r + s - m]*
     (-1)^(l - r - s)*E^(I*m*ph)*Cot[th/2]^(2*r + s - m), 
  {r, 0, l - s}], 
  Assumptions -> {Element[ph, Reals], Element[th, Reals]}

This is a direct implementation of the formula found on Wikipedia (which itself is taken from eq. (3.1) of this paper, with a different normalization).

  • $\begingroup$ Thanks. And for its conjugate, i.e. for Y bar (l,m,s), I will just replace I with -I in the above formula i.e the only change will be the term E^(-I m phi) in the above formula instead of E^(I m phi). Right? $\endgroup$
    – apk
    Commented Feb 24, 2022 at 6:53
  • $\begingroup$ I'm not sure if the relation Y bar (l,m,s) = Y(-l,m,s) holds, but you are better up applying a UpSet such as Conjugate@Y[s_, l_, m_, \[Theta]_, \[Phi]_] ^= Y[s, l, m, \[Theta], -\[Phi]]. You can also use the SpinWeightedSpherodialHarmonics package from bhptoolkit.org/users.html $\endgroup$
    – Nitaa a
    Commented Sep 21, 2023 at 13:46

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