Mismatch between definition and implementation in spherical harmonics

[ohno]

There is a mismatch between the definition and the implementation in the spherical harmonics.

$$Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.$$

Antique.jl/src/HydrogenAtom.jl

Lines 50 to 53 in 54fde9a

	function Y(model::HydrogenAtom, θ, φ; l=0, m=0)
	N = (im)^(m+abs(m)) * sqrt( (2*l+1)*factorial(l-Int(abs(m))) / (2*factorial(l+Int(abs(m)))) )
	return N * P(model,cos(θ), n=l, m=Int(abs(m))) * exp(im*m*φ) / sqrt(2*π)
	end

These are mathematically equivalent.

$$i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.$$

However, they are not equivalent programmatically.

julia> im^(abs(-1) + -1)
1 + 0im

julia> (-1)^((abs(-1) + -1)/2)
1.0

[ohno]

The phase factor is called as "Condon-Shortley Phase":

• https://en.cppreference.com/w/cpp/numeric/special_functions/assoc_legendre
• https://mathworld.wolfram.com/Condon-ShortleyPhase.html

This issue is related to #49 and #51.