Spherical harmonic conventions

The Spherical Harmonics are \begin{displaymath}Y_{lm}(\theta,\phi) = \lambda_{lm}(\cos\theta) {\rm { e}}^{{i}
m\phi}
\end{displaymath}
where the

\begin{eqnarray}\html{eqn29}
\lambda_{lm}(x) &=& \sqrt{ \frac{2l+1}{4\pi}
\fra...
...
\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{displaymath}(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{displaymath}
They are related to the ordinary Legendre Polynomials by \begin{displaymath}P_{lm} = (-1)^m (1-x^2)^{m/2} \frac{d^m}{dx^m} P_{l}(x),
\end{displaymath}
which are given by the Rodrigues formula \begin{displaymath}P_{l}(x) = \frac{1}{2^ll!}\frac{d^l}{dx^l} (x^2-1)^l.
\end{displaymath}

Note our Ylm are identical to those of Edmonds (1957) [6], even though our definition of the Plm differ by a factor (- 1)m.

Eric Hivon 2003-02-07