Spherical harmonic conventions

The Spherical Harmonics are defined as

\begin{eqnarray}
Y_{lm}(\theta,\phi) &=& \lambda_{lm}(\cos\theta) {\rm { e}}^{{i}
m\phi}
\end{eqnarray}


where the

\begin{eqnarray}
\lambda_{lm}(x) &=& \sqrt{ \frac{2l+1}{4\pi}
\frac{(l-m)!}{(l...
...
\lambda_{lm} &=& 0, \quad{\rm for}\, \vert m\vert > l.\nonumber
\end{eqnarray}


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

\begin{eqnarray}
(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} &=& 0.
\end{eqnarray}


They are related to the ordinary Legendre Polynomials Pl by

\begin{eqnarray}
P_{lm} &=& (-1)^m (1-x^2)^{m/2} \frac{d^m}{dx^m} P_{l}(x),
\end{eqnarray}


which are given by the Rodrigues formula

\begin{eqnarray}
P_{l}(x) &=& \frac{1}{2^ll!}\frac{d^l}{dx^l} (x^2-1)^l.
\end{eqnarray}


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

Eric Hivon 2005-08-31