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Spherical harmonic conventions

The Spherical Harmonics are defined as

$\displaystyle Y_{lm}(\theta,\phi)$ $\textstyle \myequal $ $\displaystyle \lambda_{lm}(\cos\theta) {\rm { e}}^{{i}
m\phi}$ (21)

where the
$\displaystyle \lambda_{lm}(x)$ $\textstyle \myequal $ $\displaystyle \sqrt{ \frac{2l+1}{4\pi}
\frac{(l-m)!}{(l+m)!} } P_{lm}(x), \quad{\rm for~}
m\ge 0$ (22)
$\displaystyle \lambda_{lm}$ $\textstyle \myequal $ $\displaystyle (-1)^m \lambda_{l\vert m\vert}, \quad{\rm for~}
m < 0,$  
$\displaystyle \lambda_{lm}$ $\textstyle \myequal $ $\displaystyle 0, \quad{\rm for}\, \vert m\vert > l.$  

Introducing $x\equiv\cos\theta$, the associated Legendre Polynomials Plm solve the differential equation

$\displaystyle (1-x^2)\frac{d^2}{dx^2}P_{lm} - 2x \frac{d}{dx}P_{lm}
+ \left(l(l+1) - \frac{m^2}{1-x^2}\right) P_{lm}$ $\textstyle \myequal $ 0. (23)

They are related to the ordinary Legendre Polynomials Pl by
Plm $\textstyle \myequal $ $\displaystyle (-1)^m (1-x^2)^{m/2} \frac{d^m}{dx^m} P_{l}(x),$ (24)

which are given by the Rodrigues formula
Pl(x) $\textstyle \myequal $ $\displaystyle \frac{1}{2^ll!}\frac{d^l}{dx^l} (x^2-1)^l.$ (25)

Note our Ylm are identical to those of [Edmonds, 1957], even though our definition of the Plm differ from his by a factor (-1)m (a.k.a. Condon-Shortley phase).

Eric Hivon 2010-06-18
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