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SphericalY

The Spherical Harmonics function

 Calling Sequence SphericalY($\mathrm{\lambda }$, $\mathrm{\mu }$, $\mathrm{\theta }$, $\mathrm{\phi }$)

Parameters

 $\mathrm{\lambda }$ - algebraic expression $\mathrm{\mu }$ - algebraic expression $\mathrm{\theta }$ - algebraic expression $\mathrm{\phi }$ - algebraic expression

Description

 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;
 $\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({{r}}^{{2}}{}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right)\right){+}\frac{\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right)\right)}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}{+}\frac{\frac{{{\partial }}^{{2}}}{{\partial }{{\mathrm{\phi }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}}{=}{0}$ (1)
 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;
 ${{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}\left({{\int }}_{{0}}^{{\mathrm{\pi }}}{\left|{\mathrm{SphericalY}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\lambda }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right|}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\phi }}{=}{1}$ (2)
 so that when written in terms of the associated LegendreP function of the first kind, SphericalY is given by
 > FunctionAdvisor( definition, SphericalY );
 $\left[{\mathrm{SphericalY}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}\frac{{\left({-1}\right)}^{{\mathrm{\mu }}}{}\sqrt{\frac{{2}{}{\mathrm{\lambda }}{+}{1}}{{\mathrm{\pi }}}}{}\sqrt{\left({\mathrm{\lambda }}{-}{\mathrm{\mu }}\right){!}}{}{{ⅇ}}^{{I}{}{\mathrm{\phi }}{}{\mathrm{\mu }}}{}{\mathrm{LegendreP}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)\right)}{{2}{}\sqrt{\left({\mathrm{\lambda }}{+}{\mathrm{\mu }}\right){!}}}{,}{¬}\left({\mathrm{\lambda }}{+}{\mathrm{\mu }}\right){::}{{ℤ}}^{{-}}{\wedge }{¬}\left({\mathrm{\lambda }}{-}{\mathrm{\mu }}\right){::}{{ℤ}}^{{-}}\right]$ (3)
 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 ${\left(-1\right)}^{\mathrm{\mu }}$. 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);
 $\frac{\sqrt{\frac{{2}{}{\mathrm{\lambda }}{+}{1}}{{\mathrm{\pi }}}}{}\sqrt{\left({\mathrm{\lambda }}{-}{\mathrm{\mu }}\right){!}}}{\sqrt{\left({\mathrm{\lambda }}{+}{\mathrm{\mu }}\right){!}}}$ (4)
 can be combined,
 > combine((4)) assuming posint;
 $\sqrt{\frac{\left({2}{}{\mathrm{\lambda }}{+}{1}\right){}\left({\mathrm{\lambda }}{-}{\mathrm{\mu }}\right){!}}{{\mathrm{\pi }}{}\left({\mathrm{\lambda }}{+}{\mathrm{\mu }}\right){!}}}$ (5)
 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];
 ${{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}\left({{\int }}_{{0}}^{{\mathrm{\pi }}}{\mathrm{SphericalY}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){}\stackrel{{&conjugate0;}}{{\mathrm{SphericalY}}{}\left({\mathrm{\rho }}{,}{\mathrm{\nu }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\phi }}{=}{{\mathrm{\delta }}}_{{\mathrm{\lambda }}{,}{\mathrm{\rho }}}{}{{\mathrm{\delta }}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}$ (6)
 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}{}{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{\mu }}{-}{\mathrm{\lambda }}\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{\mu }}{-}{\mathrm{\lambda }}\right){::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\left({-}{\mathrm{\lambda }}{-}{\mathrm{\mu }}\right){::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\left({\mathrm{\mu }}{+}{\mathrm{\lambda }}{+}{1}\right){::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\left({\mathrm{\lambda }}{-}{\mathrm{\mu }}{+}{1}\right){::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\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}{}{I}{}{\mathrm{\mu }}{}{\mathrm{\phi }}}{,}{\mathrm{\lambda }}{::}{{ℤ}}^{\left({0}{,}{+}\right)}{\wedge }{\mathrm{\mu }}{::}{ℤ}{\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({-1}\right)}^{{\mathrm{\mu }}}{}\stackrel{{&conjugate0;}}{{\mathrm{SphericalY}}{}\left({\mathrm{\lambda }}{,}{-}{\mathrm{\mu }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{,}{\mathrm{\lambda }}{::}{{ℤ}}^{\left({0}{,}{+}\right)}{\wedge }{\mathrm{\mu }}{::}{ℤ}{\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}{}{n}{}{\mathrm{\pi }}{+}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){,}{n}{::}{ℤ}\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}{::}{ℤ}{\wedge }{\mathrm{\mu }}{\ne }{0}\right]\right]$ (7)

Examples

Expressing SphericalY in terms of LegendreP

 > $\mathrm{convert}\left(\mathrm{SphericalY}\left(\mathrm{\lambda },\mathrm{\mu },\mathrm{\theta },\mathrm{\phi }\right),\mathrm{LegendreP}\right)$
 $\frac{{\left({-1}\right)}^{{\mathrm{\mu }}}{}\sqrt{\frac{{2}{}{\mathrm{\lambda }}{+}{1}}{{\mathrm{\pi }}}}{}\sqrt{\left({\mathrm{\lambda }}{-}{\mathrm{\mu }}\right){!}}{}{{ⅇ}}^{{I}{}{\mathrm{\phi }}{}{\mathrm{\mu }}}{}{\mathrm{LegendreP}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)\right)}{{2}{}\sqrt{\left({\mathrm{\lambda }}{+}{\mathrm{\mu }}\right){!}}}$ (8)

In the typical case where $\mathrm{\lambda }$ is a positive integer, $\mathrm{\mu }$ is an integer and $\left|\mathrm{\mu }\right|\le \mathrm{\lambda }$ the square roots are automatically combined resulting in the form frequently found in textbooks

 > $\mathrm{convert}\left(\mathrm{SphericalY}\left(\mathrm{\lambda },\mathrm{\mu },\mathrm{\theta },\mathrm{\phi }\right),\mathrm{LegendreP}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\lambda }::\mathrm{posint},\mathrm{\mu }::\mathrm{integer},\mathrm{abs}\left(\mathrm{\mu }\right)\le \mathrm{\lambda }$
 $\frac{{\left({-1}\right)}^{{\mathrm{\mu }}}{}\sqrt{\frac{\left({2}{}{\mathrm{\lambda }}{+}{1}\right){}\left({\mathrm{\lambda }}{-}{\mathrm{\mu }}\right){!}}{{\mathrm{\pi }}{}\left({\mathrm{\lambda }}{+}{\mathrm{\mu }}\right){!}}}{}{{ⅇ}}^{{I}{}{\mathrm{\phi }}{}{\mathrm{\mu }}}{}{\mathrm{LegendreP}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)\right)}{{2}}$ (9)

Special values

 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{SphericalY}\right)$
 $\left[\left[{\mathrm{SphericalY}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}{0}{,}{2}{}{\mathrm{\lambda }}{+}{1}{=}{0}\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}{0}{,}{\mathrm{\mu }}{::}{ℤ}{\wedge }\left(\frac{{\mathrm{\theta }}}{{\mathrm{\pi }}}\right){::}{\mathrm{even}}\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}{0}{,}{\mathrm{\Re }}{}\left({\mathrm{\mu }}\right){<}{0}{\wedge }\left(\frac{{\mathrm{\theta }}}{{\mathrm{\pi }}}\right){::}{\mathrm{even}}\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}{0}{,}{\mathrm{\lambda }}{::}{{ℤ}}^{\left({0}{,}{+}\right)}{\wedge }{\mathrm{\mu }}{::}{{ℤ}}^{{+}}{\wedge }{\mathrm{\lambda }}{<}\left|{\mathrm{\mu }}\right|\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}{0}{,}{\mathrm{\lambda }}{::}{{ℤ}}^{\left({0}{,}{+}\right)}{\wedge }\left(\frac{{\mathrm{\theta }}}{{\mathrm{\pi }}}\right){::}{ℤ}{\wedge }{\mathrm{\mu }}{::}{{ℤ}}^{{+}}\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}\frac{{\left({-1}\right)}^{\frac{{\mathrm{\lambda }}{}\left({2}{}⌊\frac{{\mathrm{\theta }}}{{2}{}{\mathrm{\pi }}}⌋{}{\mathrm{\pi }}{-}{\mathrm{\theta }}\right)}{{\mathrm{\pi }}}}{}\sqrt{\frac{{2}{}{\mathrm{\lambda }}{+}{1}}{{\mathrm{\pi }}}}}{{2}}{,}{\mathrm{\lambda }}{::}{{ℤ}}^{\left({0}{,}{+}\right)}{\wedge }\left(\frac{{\mathrm{\theta }}}{{\mathrm{\pi }}}\right){::}{ℤ}{\wedge }{\mathrm{\mu }}{=}{0}\right]\right]$ (10)

Hypergeometric representation

 > $\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{SphericalY},\mathrm{hypergeom}\right)$
 $\left[{\mathrm{SphericalY}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}\frac{{\left({-1}\right)}^{{\mathrm{\mu }}}{}\sqrt{\frac{{2}{}{\mathrm{\lambda }}{+}{1}}{{\mathrm{\pi }}}}{}\sqrt{\left({\mathrm{\lambda }}{-}{\mathrm{\mu }}\right){!}}{}{{ⅇ}}^{{I}{}{\mathrm{\phi }}{}{\mathrm{\mu }}}{}{\left({\mathrm{cos}}{}\left({\mathrm{\theta }}\right){+}{1}\right)}^{\frac{{\mathrm{\mu }}}{{2}}}{}{\mathrm{hypergeom}}{}\left(\left[{-}{\mathrm{\lambda }}{,}{\mathrm{\lambda }}{+}{1}\right]{,}\left[{1}{-}{\mathrm{\mu }}\right]{,}\frac{{1}}{{2}}{-}\frac{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}{{2}}\right)}{{2}{}\sqrt{\left({\mathrm{\lambda }}{+}{\mathrm{\mu }}\right){!}}{}{\left({\mathrm{cos}}{}\left({\mathrm{\theta }}\right){-}{1}\right)}^{\frac{{\mathrm{\mu }}}{{2}}}{}{\mathrm{\Gamma }}{}\left({1}{-}{\mathrm{\mu }}\right)}{,}{¬}\left({\mathrm{\lambda }}{+}{\mathrm{\mu }}\right){::}{{ℤ}}^{{-}}{\wedge }{¬}\left({\mathrm{\lambda }}{-}{\mathrm{\mu }}\right){::}{{ℤ}}^{{-}}{\wedge }{¬}\left({1}{-}{\mathrm{\mu }}\right){::}{{ℤ}}^{\left({0}{,}{-}\right)}\right]$ (11)

References

 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.

 See Also