#ifndef YLM_H
#define YLM_H

#include <complex>
#include <vector>

using namespace std ;

/**
* Multiple of a Spherical Harmonics.
* Edmonds, Angular momentum in quantum mechanics, 1957,
* (2.5.29): c* (-1)^m*sqrt{ (2l+1)/(4pi) (l-m)!/(l+m)!} P_l^m(cos theta) exp(i*m*phi)
* P_l^m = (1-x^2)^(m/2) d^m P_l(x)/dx^m
* int(dOmega conj(Y (l1,m1))*Y(l2,m2) = delta(l1,l2)delta(m1,m2) where dOmega=sin (theta)dtheta dphi
* (2.5.18): P_l^(-m)(x)=(-)^m (l-m)!/(l+m)! P_l^m(x)
* int dOmega Y(l1,m1) Y(l2,m2) Y(l3,m3) = sqrt{(2l1+1)(2l2+1)(2l3+1)/4/pi} j3(l1 l2 l3, 0 0 0) *j3(l1 l2 l3, m1 m2 m3)
*
* Parity w.r.t image of the argument, r->-r, phi->phi+180 deg, theta->180-theta
*  exp(i*m*phi)->exp(i*m*phi)*exp(i*m*pi)=exp(i*m*phi)*(-)^m.
*  cos(theta)-> -cos(theta), P_l^m(cos theta)-> (-)^(l-m) P_l^m(cos theta) since it
*    has parity l-m (see B.72 of the Messiah appendix)
*  full contribution of r->-r cancellation of the two (-)^m and  remains (-)^l
*  This parity is the same as in Messiah (B.91), Messiah uses the same additional
*  factor (-)^m in his definition.
*
*/
class Ylm {
public:
	/** Angular momentum quantum number. l = 0,1,2,...
	*/
	int l ;

	/** Magnetic quantum number, -l<=m<= l.
	*/
	int m ;

	/** Any additional factor or coefficient. The implicit sqrt() is an extra factor, not part of this number.
	* Although the individual Ylm is defined as a complex number field which does not specifiy any
	* additional degree of freedom, this extra parameter is useful to represent vectors of Ylm's
	* (linear combinations) with mixed expansion coeffients>
	*/
	complex<double> c ;

#if 0
	Ylm() ;
#endif

	Ylm(const int orbl, const int magm, const complex<double> coef=1.) ;

#if 0
	Ylm conj() const ;
#endif

	Ylm & operator*= (const complex<double> & fact) ;

protected:
private:
} ;

Ylm operator*(const complex<double> & fact, const Ylm & right) ;

Ylm conj(const Ylm & arg) ;

vector<Ylm> operator*(const Ylm & left, const Ylm & right) ;

bool operator== (const Ylm & first, const Ylm & secnd) ;

bool operator!= (const Ylm & first, const Ylm & secnd) ;

ostream & operator<<(ostream &os, const Ylm & someylm) ;

#endif /* YLM_H */