```/*
LEGNDR.I
Compute Legendre polynomials and associated Legendre functions.

\$Id\$
*/
/*    Copyright (c) 1994.  The Regents of the University of California.

/* After Numerical Recipes, by Press, et. al., section 6.6.  */

func legndr (l,m, x)
/* DOCUMENT legndr(l,m, x)
return the associated Legendre function Plm(x).  The X may
be an array (-1<=x<=1), but L and M (0<=M<=L) must be scalar
values.  For m=0, these are the Legendre polynomials Pl(x).
Relation of Plm(x) to Pl(x):
Plm(x) = (-1)^m (1-x^2)^(m/2) d^m/dx^m(Pl(x))
Relation of Plm(x) to spherical harmonics Ylm:
Ylm(theta,phi)= sqrt((2*l+1)(l-m)!/(4*pi*(l+m)!)) *
Plm(cos(theta)) * exp(1i*m*phi)
*/
{
m+= 0;
l+= 0;
if (structof(m)!=long || structof(l)!=long ||
m<0 || m>l) error, "l, m integers with 0<=m<=l for Plm";

if (m > 0) {
/* sqrt blows up if x out of range */
pmm= ((m%2? -1:1) * exp(sum(log(indgen(1:2*m:2))))) *
(sqrt((1.-x)*(1.+x))^m);
} else {
if (anyof(abs(x)>1.0)) error, "-1<=x<=1 for Plm";
pmm= array(1.0, dimsof(x));
}

if (l==m) return pmm;

pmmp1= (2*m+1)*x*pmm;
if (l==m+1) return pmmp1;

for (ll=m+2 ; ; ++ll) {
c1= double(ll-m);
c= (ll+m-1)/c1;
c1= (2*ll-1)/c1;
pll= c1*x*pmmp1 - c*pmm;
if (ll==l) return pll;
pmm= pmmp1;
pmmp1= pll;
}
}

func ylm_coef (l,m)
/* DOCUMENT ylm_coef(l,m)
return sqrt((2*l+1)(l-m)!/(4*pi*(l+m)!)), the normalization
coefficient for spherical harmonic Ylm with respect to the
associated Legendre function Plm.  In this implementation,
0<=m<=l; use symmetry for m<0, or use sines and cosines
instead of complex exponentials.  Unlike Plm, array L and M
arguments are permissible here.
WARNING: These get combinitorially small with large L and M;
probably Plm is simultaneously blowing up and should be
normalized directly in legndr if what you want is Ylm.  But
I don't feel like working all that out -- if you need large
L and M results, you should probably be working with some
sort of asymptotic form anyway...
*/
{
m+= 0;
l+= 0;
if (structof(m)!=long || structof(l)!=long ||
anyof(m<0) || anyof(m>l))
error, "l, m integers with 0<=m<=l; use symmetry for m<0";

lm= l-m;
not_scalar= dimsof(lm)(1);
if (not_scalar) result= array(1.0, dimsof(lm));
else result= [1.0];
iactive= indgen(numberof(lm));
for (list=where(lm) ; numberof(list) ; list=where(lm)) {
lm= lm(list);
iactive= iactive(list);
result(iactive)*= lm;
lm-= 1;
}

lm= l+m;
iactive= indgen(numberof(lm));
for (list=where(lm) ; numberof(list) ; list=where(lm)) {
lm= lm(list);
iactive= iactive(list);
result(iactive)/= lm;
lm-= 1;
}

result= sqrt(((2*l+1)/(4.*pi)) * result);
return not_scalar? result : result(1);
}
```