SphericalY($\mathrm{\lambda }$, $\mathrm{\mu }$, $\mathrm{\theta }$, $\mathrm{\phi }$) represents spherical harmonics, that is, the angular part of the solution to Laplace's equation in spherical coordinates ($r,\mathrm{\theta },\mathrm{\phi }$).

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;

$\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}f\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)+\frac{\frac{\partial }{\partial \mathrm{\theta }}\left(\sin \left(\mathrm{\theta }\right)\frac{\partial }{\partial \mathrm{\theta }}f\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)}{\sin \left(\mathrm{\theta }\right)}+\frac{\frac{{\partial }^2}{\partial {\mathrm{\phi }}^2}f\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)}{{\sin \left(\mathrm{\theta }\right)}^2}=0$

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;

${\int }_0^{2\mathrm{\pi }}{\int }_0^{\mathrm{\pi }}{\left|\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\lambda }\,\mathrm{\theta }\,\mathrm{\phi }\right)\right|}^2\sin \left(\mathrm{\theta }\right)\ⅆ\mathrm{\theta }\ⅆ\mathrm{\phi }=1$

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

FunctionAdvisor( definition, SphericalY );

$\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)=\frac{{\left(\mathrm{-1}\right)}^{\mathrm{\mu }}\sqrt{\frac{2\mathrm{\lambda }+1}{\mathrm{\pi }}}\sqrt{\left(\mathrm{\lambda }-\mathrm{\mu }\right)!}{\ⅇ}^{\mathrm{I}\mathrm{\phi }\mathrm{\mu }}\mathrm{LegendreP}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\cos \left(\mathrm{\theta }\right)\right)}{2\sqrt{\left(\mathrm{\lambda }+\mathrm{\mu }\right)!}}\,\neg \left(\mathrm{\lambda }+\mathrm{\mu }\right)::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\'\mathrm{negint}\'\right]\right)\wedge \neg \left(\mathrm{\lambda }-\mathrm{\mu }\right)::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\'\mathrm{negint}\'\right]\right)\right]$

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 ${\left(\mathrm{-1}\right)}^{\mathrm{\mu }}$. In second place, the Maple choice for the branch cuts of $\mathrm{LegendreP}\left(\mathrm{\lambda }\,\mathrm{\mu }\,z\right)$ follow conventions which, for $\mathrm{\lambda }$ and $\mathrm{\mu }$ not integers and outside a unit circle around $z=0$, are slightly different than those presented for instance in the first reference below. Finally, noting that SphericalY is more frequently used with $\mathrm{\lambda }$ and $\mathrm{\mu }$ integers, $\mathrm{\lambda }$ positive and $\left|\mathrm{\mu }\right|\le \mathrm{\lambda }$, 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);

$\frac{\sqrt{\frac{2\mathrm{\lambda }+1}{\mathrm{\pi }}}\sqrt{\left(\mathrm{\lambda }-\mathrm{\mu }\right)!}}{\sqrt{\left(\mathrm{\lambda }+\mathrm{\mu }\right)!}}$

can be combined,

combine((4)) assuming posint;

$\sqrt{\frac{\left(2\mathrm{\lambda }+1\right)\left(\mathrm{\lambda }-\mathrm{\mu }\right)!}{\mathrm{\pi }\left(\mathrm{\lambda }+\mathrm{\mu }\right)!}}$

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 $\mathrm{\lambda }$ or $\mathrm{\mu }$.

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];

${\int }_0^{2\mathrm{\pi }}{\int }_0^{\mathrm{\pi }}\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)\stackrel{\&conjugate0;}{\mathrm{SphericalY}\left(\mathrm{\rho }\,\mathrm{\nu }\,\mathrm{\theta }\,\mathrm{\phi }\right)}\sin \left(\mathrm{\theta }\right)\ⅆ\mathrm{\theta }\ⅆ\mathrm{\phi }={\mathrm{\delta }}_{\mathrm{\lambda },\mathrm{\rho }}{\mathrm{\delta }}_{\mathrm{\mu },\mathrm{\nu }}$

where in the right-hand side we have Kronecker deltas. Due to the rich structure of these functions, including periodicity with respect to both $\mathrm{\theta }$ and $\mathrm{\phi }$ 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 );

$\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)=\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,-\mathrm{\theta }\,\mathrm{\phi }\right)\,\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)=\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,-\mathrm{\phi }\right){\ⅇ}^{2\mathrm{I}\mathrm{\mu }\mathrm{\phi }}\,\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)=\frac{\mathrm{SphericalY}\left(-1-\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)\sqrt{2\mathrm{\lambda }+1}\sqrt{\mathrm{\Gamma }\left(-\mathrm{\lambda }+\mathrm{\mu }\right)}\sqrt{\mathrm{\Gamma }\left(\mathrm{\lambda }-\mathrm{\mu }+1\right)}}{\sqrt{-1-2\mathrm{\lambda }}\sqrt{\mathrm{\Gamma }\left(-\mathrm{\lambda }-\mathrm{\mu }\right)}\sqrt{\mathrm{\Gamma }\left(\mathrm{\mu }+\mathrm{\lambda }+1\right)}}\,\left(-\mathrm{\lambda }+\mathrm{\mu }\right)::\left(\neg \mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\'\mathrm{nonposint}\'\right]\right)\right)\wedge \left(-\mathrm{\lambda }-\mathrm{\mu }\right)::\left(\neg \mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\'\mathrm{nonposint}\'\right]\right)\right)\wedge \left(\mathrm{\mu }+\mathrm{\lambda }+1\right)::\left(\neg \mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\'\mathrm{nonposint}\'\right]\right)\right)\wedge \left(\mathrm{\lambda }-\mathrm{\mu }+1\right)::\left(\neg \mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\'\mathrm{nonposint}\'\right]\right)\right)\right]\,\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)=\mathrm{SphericalY}\left(\mathrm{\lambda }\,-\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right){\ⅇ}^{2\mathrm{I}\mathrm{\mu }\mathrm{\phi }}\,\mathrm{\lambda }::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\'\mathrm{nonnegint}\'\right]\right)\wedge \mathrm{\mu }::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\'\mathrm{integer}\'\right]\right)\wedge \mathrm{\mu }\le \mathrm{\lambda }\wedge -\mathrm{\lambda }\le -\mathrm{\mu }\right]\,\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)={\left(\mathrm{-1}\right)}^{\mathrm{\mu }}\stackrel{\&conjugate0;}{\mathrm{SphericalY}\left(\mathrm{\lambda }\,-\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)}\,\mathrm{\lambda }::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\'\mathrm{nonnegint}\'\right]\right)\wedge \mathrm{\mu }::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\'\mathrm{integer}\'\right]\right)\wedge \mathrm{\mu }\le \mathrm{\lambda }\wedge -\mathrm{\lambda }\le -\mathrm{\mu }\right]\,\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)=\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,2\mathrm{\pi }n+\mathrm{\theta }\,\mathrm{\phi }\right)\,n::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\'\mathrm{integer}\'\right]\right)\right]\,\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)=\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }+\frac{2\mathrm{\pi }n}{\mathrm{\mu }}\right)\,n::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\'\mathrm{integer}\'\right]\right)\wedge \mathrm{\mu }\ne 0\right]\right]$

$\mathrm{convert}\left(\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)\,\mathrm{LegendreP}\right)$

$\frac{{\left(\mathrm{-1}\right)}^{\mathrm{\mu }}\sqrt{\frac{2\mathrm{\lambda }+1}{\mathrm{\pi }}}\sqrt{\left(\mathrm{\lambda }-\mathrm{\mu }\right)!}{\ⅇ}^{\mathrm{I}\mathrm{\phi }\mathrm{\mu }}\mathrm{LegendreP}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\cos \left(\mathrm{\theta }\right)\right)}{2\sqrt{\left(\mathrm{\lambda }+\mathrm{\mu }\right)!}}$

$\mathrm{convert}\left(\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)\,\mathrm{LegendreP}\right)\mathbf{assuming}\mathrm{\lambda }::\'\mathrm{posint}\',\mathrm{\mu }::\'\mathrm{integer}\',\mathrm{abs}\left(\mathrm{\mu }\right)\le \mathrm{\lambda }$

$\frac{{\left(\mathrm{-1}\right)}^{\mathrm{\mu }}\sqrt{\frac{\left(2\mathrm{\lambda }+1\right)\left(\mathrm{\lambda }-\mathrm{\mu }\right)!}{\mathrm{\pi }\left(\mathrm{\lambda }+\mathrm{\mu }\right)!}}{\ⅇ}^{\mathrm{I}\mathrm{\phi }\mathrm{\mu }}\mathrm{LegendreP}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\cos \left(\mathrm{\theta }\right)\right)}{2}$

$\mathrm{FunctionAdvisor}\left(\mathrm{special\_values}\,\mathrm{SphericalY}\right)$

$\left[\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\&comma;\mathrm{\mu }\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\right)=0\&comma;2\mathrm{\lambda }+1=0\right]\&comma;\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\&comma;\mathrm{\mu }\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\right)=0\&comma;\mathrm{\mu }::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\&apos;\mathrm{integer}\&apos;\right]\right)\wedge \left(\frac{\mathrm{\theta }}{\mathrm{\pi }}\right)::\mathrm{even}\right]\&comma;\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\&comma;\mathrm{\mu }\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\right)=0\&comma;\mathrm{\Re }\left(\mathrm{\mu }\right)<0\wedge \left(\frac{\mathrm{\theta }}{\mathrm{\pi }}\right)::\mathrm{even}\right]\&comma;\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\&comma;\mathrm{\mu }\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\right)=0\&comma;\mathrm{\lambda }::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\&apos;\mathrm{nonnegint}\&apos;\right]\right)\wedge \mathrm{\mu }::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\&apos;\mathrm{posint}\&apos;\right]\right)\wedge \mathrm{\lambda }<\left|\mathrm{\mu }\right|\right]\&comma;\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\&comma;\mathrm{\mu }\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\right)=0\&comma;\mathrm{\lambda }::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\&apos;\mathrm{nonnegint}\&apos;\right]\right)\wedge \left(\frac{\mathrm{\theta }}{\mathrm{\pi }}\right)::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\&apos;\mathrm{integer}\&apos;\right]\right)\wedge \mathrm{\mu }::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\&apos;\mathrm{posint}\&apos;\right]\right)\right]\&comma;\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\&comma;\mathrm{\mu }\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\right)=\frac{{\left(\mathrm{-1}\right)}^{\frac{\mathrm{\lambda }\left(2⌊\frac{\mathrm{\theta }}{2\mathrm{\pi }}⌋\mathrm{\pi }-\mathrm{\theta }\right)}{\mathrm{\pi }}}\sqrt{\frac{2\mathrm{\lambda }+1}{\mathrm{\pi }}}}{2}\&comma;\mathrm{\lambda }::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\&apos;\mathrm{nonnegint}\&apos;\right]\right)\wedge \left(\frac{\mathrm{\theta }}{\mathrm{\pi }}\right)::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\&apos;\mathrm{integer}\&apos;\right]\right)\wedge \mathrm{\mu }=0\right]\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{specialize}\&comma;\mathrm{SphericalY}\&comma;\mathrm{hypergeom}\right)$

$\left[\mathrm{SphericalY}\left(\mathrm{\lambda }\&comma;\mathrm{\mu }\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\right)=\frac{{\left(\mathrm{-1}\right)}^{\mathrm{\mu }}\sqrt{\frac{2\mathrm{\lambda }+1}{\mathrm{\pi }}}\sqrt{\left(\mathrm{\lambda }-\mathrm{\mu }\right)!}{\&ExponentialE;}^{\mathrm{I}\mathrm{\phi }\mathrm{\mu }}{\left(\cos \left(\mathrm{\theta }\right)+1\right)}^{\frac{\mathrm{\mu }}{2}}\mathrm{hypergeom}\left(\left[-\mathrm{\lambda }\&comma;\mathrm{\lambda }+1\right]\&comma;\left[1-\mathrm{\mu }\right]\&comma;\frac{1}{2}-\frac{\cos \left(\mathrm{\theta }\right)}{2}\right)}{2\sqrt{\left(\mathrm{\lambda }+\mathrm{\mu }\right)!}{\left(\cos \left(\mathrm{\theta }\right)-1\right)}^{\frac{\mathrm{\mu }}{2}}\mathrm{\Gamma }\left(1-\mathrm{\mu }\right)}\&comma;\neg \left(\mathrm{\lambda }+\mathrm{\mu }\right)::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\&apos;\mathrm{negint}\&apos;\right]\right)\wedge \neg \left(\mathrm{\lambda }-\mathrm{\mu }\right)::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\&apos;\mathrm{negint}\&apos;\right]\right)\wedge \neg \left(1-\mathrm{\mu }\right)::\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\&apos;\mathrm{nonposint}\&apos;\right]\right)\right]$

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.