Re: [Help-gsl] spherical harmonics for m<0 (m=-l)

[John Gehman]

There is a relationship -- look it up for sure, obviously, but I  
believe the negative values of m are simply the complex conjugate  
(relevant in the exponential) of the positive m values, and maybe the  
m = \pm 1 are opposite overall sign to each other.

I hesitated to reply with this earlier, as I presumed that mucking  
into the functions with this degree of detail to solve the problem  
defeats the purpose of a simple gsl function, but in light of Brian's  
reply, perhaps it helps ... ?

Cheers
john

On 28/09/2005, at 2:44 AM, Brian Gough wrote:

> Drew Parsons writes:
>
> > I'm working with spherical harmonics, calculated a value for each l
> > separately by putting together a sum over m of Y_l^m (averaging the
> > value of the spherical harmonic over a number of neighbouring  
> > points in
> > space) , as in
> >
> > \sum_{m=-l}^{l}  < Y_l^m (\theta, \phi ) >
> >
> > To help get this done GSL offers me gsl_sf_legendre_sphPlm( l, m,  
> > x ),
> > but the function only accepts m >= 0.
> >
> > What is the best way to proceed to also count the cases where m < 0 ?
>
> I think there is a relationship between +m and -m (Abramowitz &Stegun
> 8.2.5)
>
> If you are computing multiple values you'll want to use the
> sphPlm_array function for efficiency.
>
> I'm not sure why the original function is restricted to m>=0, maybe
> there was a reason for that.
>
> --
> Brian Gough
>
> Network Theory Ltd,
> Publishing Free Software Manuals --- http://www.network-theory.co.uk/
>
>
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