I am trying to expand in spherical harmonics the following real function:
$f(\theta, \phi) = \sin{\theta} \left| \cos{\phi} \right|$
by applying the definition of the spherical harmonics expansion it is possible to write
$f(\theta, \phi) = \sum_{k=0}^{\infty}\sum_{m=-k}^{k} f_{k \, m} Y_{k \, m}^{*}$
where $Y_{k \, m}^{*} $ are spherical harmonics.
The coefficients $f_{k \, m}$ are then given by:
$f_{k \, m}= \int_{0}^{\pi}\int_{0}^{2\pi} f(\theta, \phi) \, \sin{\theta}\, Y_{k \, m}^{*}\, d\phi \, d\theta$
with $Y_{k \, m}^{*}$
until now here everything is fine. However, when I carry out the integration above (with mathematica) to compute the coefficients $f_{k \, m}$ i found that $\, f_{k \, m} = f_{k \, -m}$.
But, in the following paper the authors are expanding exactly the same function: https://doi.org/10.1017/jfm.2015.129 (Pag 11 and 12; Eq 3.5-3.9)
In fact, the authors claim that the terms $f_{k \, m}$ with $m<0$ vanish.
I am abit clueless, despite the expansion step seems straightforward to me, I am not sure I am doing it correctly.
(If you can't access the paper here's a screenshot of that part):
Derivation by the authors:
[...]
You're right, it does look like an error. What they appear to be doing is using $e^{\pm im\phi} = \cos{m\phi}\pm i\sin{\phi}$ to expand in terms of sines and cosines for $m\geq 0$ instead of $e^{im\theta}$ for both positive and negative $m$. After all, $\cos{(-m\phi)}=\cos{m\phi}$ and $P^{-m}_k(x) \propto P^m_k(x)$, so the integral with a negative $m$ is proportional to that with a positive $m$. The justification should be that $P_k^m(\cos{\theta})\cos{m\phi}$ and $P_k^m(\cos{\theta})\sin{m\phi}$ for $0 \leq m \leq k$ form a basis, just as $P_k^m(\cos{\theta})e^{im\phi}$ for $-k \leq m \leq k$ do.