Wikipedia (2026)

Wikipedia is not a primary source, but it is probably the first place most people look, so it is worth recording exactly how its conventions relate to ours. The relevant articles are Wigner D-matrix, Spin-weighted spherical harmonics, Spherical harmonics, and Euler angles. All quotations below were taken from the versions of those pages current on 2026-09-08 (the Wigner D-matrix article was last edited 2026-05-04 and the spin-weighted spherical harmonics article 2026-02-09 at that time). Wikipedia articles change, so a later reader should expect to re-check them.

Euler angles

The Euler angles article describes our convention as the "proper Euler angles" in the $z$-$y$-$z$ (or $Z$-$Y'$-$Z''$) form, and notes that an extrinsic sequence of rotations about fixed axes is equivalent to an intrinsic sequence about the moving axes taken in the reverse order — which is exactly the observation that lets us write $𝐑_{α,β,γ} = e^{α𝐤/2} e^{β𝐣/2} e^{γ𝐤/2}$ either way. There is nothing to test here.

Wigner $D$-matrix

The Wigner D-matrix article defines the rotation operator

\[\mathcal{R}(α, β, γ) = e^{-iα J_z} e^{-iβ J_y} e^{-iγ J_z},\]

and then

\[D^j_{m'm}(α, β, γ) \equiv \langle j m' | \mathcal{R}(α, β, γ) | j m \rangle = e^{-im'α}\, d^j_{m'm}(β)\, e^{-imγ},\]

which is precisely our definition. The article gives Wigner's formula for the $d$-matrix as

\[d^j_{m'm}(β) = \left[(j+m')!\,(j-m')!\,(j+m)!\,(j-m)!\right]^{1/2} \sum_{s=s_{\min}}^{s_{\max}} \left[ \frac{(-1)^{m'-m+s} \left(\cos\frac{β}{2}\right)^{2j+m-m'-2s} \left(\sin\frac{β}{2}\right)^{m'-m+2s}} {(j+m-s)!\, s!\, (m'-m+s)!\, (j-m'-s)!} \right],\]

with $s_{\min} = \max(0, m-m')$ and $s_{\max} = \min(j+m, j-m')$, and lists the explicit elements for $j = 1/2, 1, 2$, of which we transcribe those for $j = 1$ and $j = 2$. It records the symmetries

\[D^j_{m'm}(α, β, γ) = (-1)^{m'-m}\, \overline{D^j_{-m',-m}(α, β, γ)}, \qquad d^j_{m',m}(-β) = d^j_{m,m'}(β) = (-1)^{m'-m}\, d^j_{m',m}(β),\]

and the relation to the spherical harmonics

\[D^{ℓ}_{m0}(α, β, γ) = \sqrt{\frac{4π}{2ℓ+1}}\; \overline{Y_{ℓ}^{m}(β, α)},\]

all of which agree with ours.

The section "Properties of the Wigner D-matrix" introduces "space-fixed" operators $\hat{\mathcal{J}}$ and "body-fixed" operators $\hat{\mathcal{P}}$ in terms of Euler-angle derivatives, with

\[\hat{\mathcal{J}}_3 = -i \frac{\partial}{\partial α}, \qquad \hat{\mathcal{P}}_3 = -i \frac{\partial}{\partial γ},\]

and states that it is the complex conjugate $D^{j\ast}_{m'm}$ that is the simultaneous eigenfunction, with $\mathcal{J}_3 D^{j\ast}_{m'm} = m' D^{j\ast}_{m'm}$ and $\mathcal{P}_3 D^{j\ast}_{m'm} = m D^{j\ast}_{m'm}$. Comparing with our operators, $\hat{\mathcal{J}}_3 = L_z$ while $\hat{\mathcal{P}}_3 = -R_z$; more generally $\mathcal{P} = -R$ (see the calculation page). Conjugating Wikipedia's eigenvalue statements gives $L_z D_{m'm} = -m' D_{m'm}$ and $R_z D_{m'm} = m D_{m'm}$, which is exactly what we find.

Spherical harmonics

The Spherical harmonics article uses standard physicists' coordinates ($θ$ the colatitude, $ϕ$ the azimuth), and gives the quantum-mechanical definition, including the Condon–Shortley phase, as

\[Y_ℓ^m(θ, ϕ) = (-1)^m \sqrt{\frac{2ℓ+1}{4π} \frac{(ℓ-m)!}{(ℓ+m)!}}\; \tilde{P}_ℓ^m(\cos θ)\, e^{imϕ},\]

where $\tilde{P}_ℓ^m$ is the associated Legendre function without the Condon–Shortley phase,

\[\tilde{P}_ℓ^m(x) = (1-x^2)^{m/2} \frac{d^m}{dx^m} P_ℓ(x), \qquad P_ℓ(x) = \frac{1}{2^ℓ ℓ!} \frac{d^ℓ}{dx^ℓ} (x^2-1)^ℓ,\]

so as to "avoid counting the phase twice"; and

\[\overline{Y_ℓ^m(θ, ϕ)} = (-1)^m\, Y_ℓ^{-m}(θ, ϕ).\]

We implement this with the derivatives evaluated by automatic differentiation, exactly as on the Condon–Shortley page.

Spin-weighted spherical harmonics

The Spin-weighted spherical harmonics article follows Newman and Penrose: a quantity of spin weight $s$ picks up $e^{isθ}$ when the tangent basis is rotated by $θ$, and the operators

\[\eth \eta = -(\sin θ)^s \left\{ \frac{\partial}{\partial θ} + \frac{i}{\sin θ} \frac{\partial}{\partial ϕ} \right\} \left[ (\sin θ)^{-s} \eta \right], \qquad \bar{\eth} \eta = -(\sin θ)^{-s} \left\{ \frac{\partial}{\partial θ} - \frac{i}{\sin θ} \frac{\partial}{\partial ϕ} \right\} \left[ (\sin θ)^{s} \eta \right],\]

act on the harmonics as

\[\eth\, {}_sY_{ℓm} = +\sqrt{(ℓ-s)(ℓ+s+1)}\, {}_{s+1}Y_{ℓm}, \qquad \bar{\eth}\, {}_sY_{ℓm} = -\sqrt{(ℓ+s)(ℓ-s+1)}\, {}_{s-1}Y_{ℓm},\]

which are the relations tested on the Newman–Penrose page. The article gives the explicit formula

\[{}_sY_{ℓm}(θ, ϕ) = (-1)^{ℓ+m-s} \sqrt{\frac{(ℓ+m)!\,(ℓ-m)!\,(2ℓ+1)}{4π\,(ℓ+s)!\,(ℓ-s)!}}\; \sin^{2ℓ}\left(\frac{θ}{2}\right) e^{imϕ} \sum_r (-1)^r \binom{ℓ-s}{r} \binom{ℓ+s}{r+s-m} \cot^{2r+s-m}\left(\frac{θ}{2}\right),\]

the relation to the Wigner $D$-matrix

\[D^ℓ_{-m\,s}(ϕ, θ, -ψ) = (-1)^m \sqrt{\frac{4π}{2ℓ+1}}\; {}_sY_{ℓm}(θ, ϕ)\, e^{isψ},\]

and the conjugation relation $\overline{{}_sY_{ℓm}} = (-1)^{s+m}\, {}_{-s}Y_{ℓ,-m}$. Note that the explicit formula differs from Goldberg et al.'s Eq. (3.1) by exactly the factor $(-1)^m$, so we expect it to agree with ours; the $D$-matrix relation is the integer-index form of our definition, ${}_sY_{ℓ,m} = (-1)^m \sqrt{(2ℓ+1)/4π}\; 𝔇^{(ℓ)}_{-m,s}$, extended to $γ = -ψ$.

Implementing formulas

We begin by writing code that implements the formulas from Wikipedia. We encapsulate the formulas in a module so that we can test them against the SphericalFunctions package.

module Wikipedia

We'll use some predefined utilities to make the code look more like the equations.

import ..ConventionsUtilities: 𝒾, ❗, ∂ⁿ

Wigner's formula for the $d$-matrix, as quoted above:

function d(j, m′, m, β::T) where {T<:Real}    √T((j+m′)❗ * (j-m′)❗ * (j+m)❗ * (j-m)❗) *    sum(        (-1)^(m′-m+s) * cos(β/2)^(2j+m-m′-2s) * sin(β/2)^(m′-m+2s)        / T((j+m-s)❗ * (s)❗ * (m′-m+s)❗ * (j-m′-s)❗)        for s ∈ max(0, m-m′):min(j+m, j-m′);        init=zero(T)    )end

The $D$-matrix, as the matrix element of $\mathcal{R}(α, β, γ)$:

function D(j, m′, m, α, β, γ)    exp(-𝒾 * m′ * α) * d(j, m′, m, β) * exp(-𝒾 * m * γ)end

The explicit elements listed in the article for $j=1$ and $j=2$ (we transcribe the rows with $m' \geq |m|$; the others follow from the symmetries also given in the article, which we apply here):

function d_explicit(j, m′, m, β)    if abs(m′) < abs(m)        return (-1)^(m-m′) * d_explicit(j, m, m′, β)    end    if m′ < 0        return (-1)^(m-m′) * d_explicit(j, -m′, -m, β)    end    if (j, m′, m) == (1, 1, 1)        (1 + cos(β)) / 2    elseif (j, m′, m) == (1, 1, 0)        -sin(β) / √2    elseif (j, m′, m) == (1, 1, -1)        (1 - cos(β)) / 2    elseif (j, m′, m) == (1, 0, 0)        cos(β)    elseif (j, m′, m) == (2, 2, 2)        (1 + cos(β))^2 / 4    elseif (j, m′, m) == (2, 2, 1)        -sin(β) * (1 + cos(β)) / 2    elseif (j, m′, m) == (2, 2, 0)        √(3/8) * sin(β)^2    elseif (j, m′, m) == (2, 2, -1)        -sin(β) * (1 - cos(β)) / 2    elseif (j, m′, m) == (2, 2, -2)        (1 - cos(β))^2 / 4    elseif (j, m′, m) == (2, 1, 1)        (2cos(β)^2 + cos(β) - 1) / 2    elseif (j, m′, m) == (2, 1, 0)        -√(3/8) * sin(2β)    elseif (j, m′, m) == (2, 1, -1)        (-2cos(β)^2 + cos(β) + 1) / 2    elseif (j, m′, m) == (2, 0, 0)        (3cos(β)^2 - 1) / 2    else        error("No explicit formula transcribed for (j, m′, m) = ($j, $m′, $m)")    endend

The Legendre polynomial and the associated Legendre function (without Condon–Shortley phase), transcribed literally from Rodrigues' formula using the ∂ⁿ utility, which computes derivatives symbolically:

function P(ℓ, x)    1 / (2^ℓ * (ℓ)❗) * ∂ⁿ(x -> (x^2 - 1)^ℓ, ℓ)(x)endfunction P̃(ℓ, m, x)    (1 - x^2)^(m/2) * ∂ⁿ(x -> P(ℓ, x), m)(x)end

The spherical harmonics, for $m \geq 0$ from the definition, and for $m < 0$ from the conjugation relation:

function Y(ℓ, m, θ::T, ϕ::T) where {T<:Real}    if m < 0        return (-1)^m * conj(Y(ℓ, -m, θ, ϕ))    end    (-1)^m * √T((2ℓ+1) * (ℓ-m)❗ / (4big(π) * (ℓ+m)❗)) * T(P̃(ℓ, m, cos(θ))) * exp(𝒾 * m * ϕ)end

The explicit formula for the spin-weighted spherical harmonics. The sum runs over all $r$ for which the binomial coefficients are nonzero. The cotangent is singular at the north pole, so we will avoid the poles when testing this expression.

function ₛYₗₘ(s, ℓ, m, θ::T, ϕ::T) where {T<:Real}    (-1)^(ℓ+m-s) * √T((ℓ+m)❗ * (ℓ-m)❗ * (2ℓ+1) / (4big(π) * (ℓ+s)❗ * (ℓ-s)❗)) *    sin(θ/2)^(2ℓ) * exp(𝒾 * m * ϕ) *    sum(        (-1)^r * T(binomial(big(ℓ-s), r) * binomial(big(ℓ+s), r+s-m)) * cot(θ/2)^(2r+s-m)        for r ∈ max(0, m-s):min(ℓ-s, ℓ+m);        init=zero(Complex{T})    )end

end  # module Wikipedia

Tests

We can now test the functions against the equivalent functions from the SphericalFunctions package. We will need to test approximate floating-point equality, so we set absolute and relative tolerances (respectively) in terms of the machine epsilon:

ϵₐ = 100eps()ϵᵣ = 1000eps()

We only test up to

ℓₘₐₓ = 6

and

sₘₐₓ = 2

because the symbolic derivatives in the formulas become expensive to compute at higher orders, and this will be sufficient to sort out any sign or normalization differences, which are the most likely source of error. To keep the running time down, we also use modest grids of points. The explicit formula for ${}_sY_{ℓm}$ involves $\cot(θ/2)$, so we avoid the poles by a small amount.

θϕs = θϕrange(rng, Float64, 7; avoid_poles=1e-3)αβγs = αβγrange(rng, Float64, 5)

We will also need the imaginary unit from the utilities module.

import .ConventionsUtilities: 𝒾

First, Wikipedia's explicit $d$ elements agree with Wikipedia's general formula, and both agree with ours:

for β ∈ βrange(rng, Float64, 15)    for (j, m′, m) ∈ ℓm′mrange(2)        j == 0 && continue  # the article lists no j=0 element (it is just 1)        @test Wikipedia.d_explicit(j, m′, m, β) ≈ Wikipedia.d(j, m′, m, β) atol=ϵₐ rtol=ϵᵣ    end    for (j, dʲ) ∈ SphericalFunctions.dCalculator(β, ℓₘₐₓ), m′ ∈ -j:j, m ∈ -j:j        @test Wikipedia.d(j, m′, m, β) ≈ dʲ[m′, m] atol=ϵₐ rtol=ϵᵣ    endend

The $D$-matrix agrees with ours, and satisfies the conjugation symmetry quoted above:

for (α, β, γ) ∈ αβγs    for (j, 𝔇ʲ) ∈ SphericalFunctions.DCalculator(α, β, γ, ℓₘₐₓ), m′ ∈ -j:j, m ∈ -j:j        @test Wikipedia.D(j, m′, m, α, β, γ) ≈ 𝔇ʲ[m′, m] atol=ϵₐ rtol=ϵᵣ        @test Wikipedia.D(j, m′, m, α, β, γ) ≈ (-1)^(m′-m) * conj(Wikipedia.D(j, -m′, -m, α, β, γ)) atol=ϵₐ rtol=ϵᵣ    endend

The spherical harmonics agree with ours, and satisfy Wikipedia's relation to $D$:

for (θ, ϕ) ∈ θϕrange(rng, Float64, 7)    for (ℓ, Yˡ) ∈ SphericalFunctions.YlmCalculator(θ, ϕ, ℓₘₐₓ), m ∈ -ℓ:ℓ        @test Wikipedia.Y(ℓ, m, θ, ϕ) ≈ Yˡ[m] atol=ϵₐ rtol=ϵᵣ        @test Wikipedia.D(ℓ, m, 0, ϕ, θ, zero(θ)) ≈ √(4π/(2ℓ+1)) * conj(Wikipedia.Y(ℓ, m, θ, ϕ)) atol=ϵₐ rtol=ϵᵣ    endend

Finally, the spin-weighted spherical harmonics agree with ours, and satisfy Wikipedia's relation to the $D$-matrix and its conjugation symmetry:

for (θ, ϕ) ∈ θϕs    for (ℓ, Yˡ) ∈ SphericalFunctions.sYlmCalculator(θ, ϕ, ℓₘₐₓ, -sₘₐₓ:sₘₐₓ)        for s ∈ -min(ℓ, sₘₐₓ):min(ℓ, sₘₐₓ), m ∈ -ℓ:ℓ            @test Wikipedia.ₛYₗₘ(s, ℓ, m, θ, ϕ) ≈ Yˡ[s, m] atol=ϵₐ rtol=ϵᵣ            @test conj(Wikipedia.ₛYₗₘ(s, ℓ, m, θ, ϕ)) ≈ (-1)^(s+m) * Wikipedia.ₛYₗₘ(-s, ℓ, -m, θ, ϕ) atol=ϵₐ rtol=ϵᵣ            for ψ ∈ (0.0, 0.7, 2.9)                @test Wikipedia.D(ℓ, -m, s, ϕ, θ, -ψ) ≈                    (-1)^m * √(4π/(2ℓ+1)) * Wikipedia.ₛYₗₘ(s, ℓ, m, θ, ϕ) * exp(𝒾 * s * ψ) atol=ϵₐ rtol=ϵᵣ            end        end    endend

These successful tests show that Wikipedia's $d$, $D$, $Y_ℓ^m$, and ${}_sY_{ℓm}$ all agree with the corresponding functions defined by the SphericalFunctions package.

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