Correct docs for sht

Ugly, but closes #40.

Author: MikaelSlevinsky

src/SphericalHarmonics/SphericalHarmonics.jl

@@ -32,10 +32,10 @@ doc"""
 Computes the bivariate Fourier series given by the spherical harmonic expansion:
 
 ```math
-{\rm SHT} : \sum_{\ell=0}^n\sum_{m=-\ell}^{\ell} f_{\ell}^m Y_{\ell}^m(\theta,\varphi) \to \sum_{\ell=0}^n\sum_{m=-n}^{n} g_{\ell}^m \frac{e^{{\rm i} m \varphi}}{\sqrt{2\pi}} \left\{\begin{array}{c}\cos\ell\theta\\ \sin(\ell+1)\theta\end{array}\right\},
+{\rm SHT} : \sum_{\ell=0}^n\sum_{m=-\ell}^{\ell} f_{\ell}^m Y_{\ell}^m(\theta,\varphi) \to \sum_{\ell=0}^n\sum_{m=-n}^{n} g_{\ell}^m \left\{\begin{array}{ccc}\cos\ell\theta & {\rm for} & m{\rm~even}\\ \sin(\ell+1)\theta & {\rm for} & m{\rm~odd}\end{array}\right\}\times \sqrt{\frac{2-\delta_{m,0}}{\pi}} \left\{\begin{array}{ccc} \cos m\varphi & {\rm for} & m \ge 0,\\ \sin(-m\varphi) & {\rm for} & m < 0.\end{array}\right.
 ```
 
-where the cosines are used when ``m`` is even and the sines are used when ``m`` is odd. The spherical harmonic expansion coefficients are organized as follows:
+The spherical harmonic expansion coefficients are organized as follows:
 
 ```math
 F = \begin{pmatrix}
@@ -66,10 +66,10 @@ doc"""
 Computes the spherical harmonic expansion given by the bivariate Fourier series:
 
 ```math
-{\rm iSHT} : \sum_{\ell=0}^n\sum_{m=-n}^{n} g_{\ell}^m \frac{e^{{\rm i} m \varphi}}{\sqrt{2\pi}} \left\{\begin{array}{c}\cos\ell\theta\\ \sin(\ell+1)\theta\end{array}\right\} \to \sum_{\ell=0}^n\sum_{m=-\ell}^{\ell} f_{\ell}^m Y_{\ell}^m(\theta,\varphi),
+{\rm iSHT} : \sum_{\ell=0}^n\sum_{m=-n}^{n} g_{\ell}^m \left\{\begin{array}{ccc}\cos\ell\theta & {\rm for} & m{\rm~even}\\ \sin(\ell+1)\theta & {\rm for} & m{\rm~odd}\end{array}\right\}\times \sqrt{\frac{2-\delta_{m,0}}{\pi}} \left\{\begin{array}{ccc} \cos m\varphi & {\rm for} & m \ge 0,\\ \sin(-m\varphi) & {\rm for} & m < 0,\end{array}\right. \to \sum_{\ell=0}^n\sum_{m=-\ell}^{\ell} f_{\ell}^m Y_{\ell}^m(\theta,\varphi).
 ```
 
-where the cosines are used when ``m`` is even and the sines are used when ``m`` is odd. The spherical harmonic expansion coefficients are organized as follows:
+The spherical harmonic expansion coefficients are organized as follows:
 
 ```math
 F = \begin{pmatrix}

src/SphericalHarmonics/sphfunctions.jl

@@ -87,15 +87,15 @@ function maxcolnorm(A::AbstractMatrix)
 end
 
 doc"""
-Pointwise evaluation of spherical harmonic:
+Pointwise evaluation of real orthonormal spherical harmonic:
 
 ```math
-Y_\ell^m(\theta,\varphi) = \frac{e^{{\rm i} m\varphi}}{\sqrt{2\pi}} {\rm i}^{m+|m|}\sqrt{(\ell+\frac{1}{2})\frac{(\ell-m)!}{(\ell+m)!}} P_\ell^m(\cos\theta).
+Y_\ell^m(\theta,\varphi) = (-1)^{|m|}\sqrt{(\ell+\frac{1}{2})\frac{(\ell-|m|)!}{(\ell+|m|)!}} P_\ell^{|m|}(\cos\theta) \sqrt{\frac{2-\delta_{m,0}}{\pi}} \left\{\begin{array}{ccc} \cos m\varphi & {\rm for} & m \ge 0,\\ \sin(-m\varphi) & {\rm for} & m < 0.\end{array}\right.
 ```
 """
 sphevaluate(θ, φ, L, M) = sphevaluatepi(θ/π, φ/π, L, M)
 
-sphevaluatepi(θ::Number, φ::Number, L::Integer, M::Integer) = sphevaluatepi(θ,L,M)*sphevaluatepi(φ,M)
+sphevaluatepi(θ::Number, φ::Number, L::Integer, M::Integer) = sphevaluatepi(θ, L, M)*sphevaluatepi(φ, M)
 
 function sphevaluatepi(θ::Number, L::Integer, M::Integer)
     ret = one(θ)/sqrt(two(θ))
@@ -120,4 +120,4 @@ function sphevaluatepi(θ::Number, L::Integer, M::Integer)
     end
 end
 
-sphevaluatepi(φ::Number, M::Integer) = complex(cospi(M*φ),sinpi(M*φ))/sqrt(two(φ)*π)
+sphevaluatepi(φ::Number, M::Integer) = sqrt((two(φ)-δ(M, 0))/π)*(M ≥ 0 ? cospi(M*φ) : sinpi(-M*φ))