SphericalY(, , , ) represents spherical harmonics, that is, the angular part of the solution to Laplace's equation in spherical coordinates ().

Diff(r^2*Diff(f(r,theta,phi),r),r) + 1/sin(theta)*Diff(sin(theta)*Diff(f(r,theta,phi),theta),theta) + 1/sin(theta)^2*Diff(f(r,theta,phi),phi,phi) = 0;

The SphericalY functions are particularly relevant in quantum mechanics, where they are eigenfunctions of observable operators associated with angular momentum - see Abramowitz and Stegun, Chapter VI. SphericalY is normalized such that

Int(Int(abs(SphericalY(lambda,lambda,theta,phi))^2*sin(theta),theta=0..Pi),phi=0..2*Pi) = 1;

so that when written in terms of the associated LegendreP function of the first kind, SphericalY is given by

FunctionAdvisor( definition, SphericalY );

Attention should be paid to the normalization conventions adopted. The requirement that the double integral mentioned is equal to one does not fix a phase, which can then be chosen in different ways; following the definitions given by references 2 and 3 (at the bottom), thus, in Maple the right-hand side of the definition above includes the multiplicative factor . In second place, the Maple choice for the branch cuts of  follow conventions which, for  and  not integers and outside a unit circle around , are slightly different than those presented for instance in the first reference below. Finally, noting that SphericalY is more frequently used with  and  integers,  positive and , in this case the three square roots entering the definition above,

((2*lambda+1)/Pi)^(1/2)*(lambda-mu)!^(1/2)/(lambda+mu)!^(1/2);

can be combined,

combine((4)) assuming posint;

resulting into a form of the definition usually presented in textbooks - this combination of the radicals, however, is not valid for arbitrary complex values of  or .

The SphericalY functions constitute a complete set of orthonormal functions satisfying

Int(Int(SphericalY(lambda,mu,theta,phi)*conjugate(SphericalY(rho,nu,theta,phi))*sin(theta),theta=0..Pi),phi=0..2*Pi) = delta[lambda,rho]*delta[mu,nu];

where in the right-hand side we have Kronecker deltas. Due to the rich structure of these functions, including periodicity with respect to both  and  and reflection properties regarding each of its four arguments, the number of identities they satisfy is rather large. Some important ones are

FunctionAdvisor( identities, SphericalY );

Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover Publications.

Arfken, G., and Weber, H.J. Mathematical Methods for Physicists. 3rd ed. Academic Press, 1985.

Cohen-Tannoudji, C.; Diu, B.; and Laloe, F. Quantum Mechanics. Paris: Hermann, 1977. Vol. 1, Complement A-VI.