SphericalY - Maple Programming Help

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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;
 $\frac{{\partial }}{{\partial }{r}}{}\left({{r}}^{{2}}{}\left(\frac{{\partial }}{{\partial }{r}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right)\right)\right){+}\frac{\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}\left({\mathrm{sin}}{}\left({\mathrm{θ}}\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right)\right)\right)}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}{+}\frac{\frac{{{\partial }}^{{2}}}{{\partial }{{\mathrm{φ}}}^{{2}}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right)}{{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}}{=}{0}$ (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;
 ${{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{{\mathrm{π}}}{\left|{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{λ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right)\right|}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{φ}}{=}{1}$ (2)
 so that when written in terms of the associated LegendreP function of the first kind, SphericalY is given by
 $\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}\frac{{1}}{{2}}{}\frac{{\left({-}{1}\right)}^{{\mathrm{μ}}}{}\sqrt{\frac{{2}{}{\mathrm{λ}}{+}{1}}{{\mathrm{π}}}}{}\sqrt{\left({\mathrm{λ}}{-}{\mathrm{μ}}\right){!}}{}{{ⅇ}}^{{I}{}{\mathrm{φ}}{}{\mathrm{μ}}}{}{\mathrm{LegendreP}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{cos}}{}\left({\mathrm{θ}}\right)\right)}{\sqrt{\left({\mathrm{λ}}{+}{\mathrm{μ}}\right){!}}}{,}{\mathrm{And}}{}\left({\mathrm{Not}}{}\left(\left({\mathrm{λ}}{+}{\mathrm{μ}}\right){::}{\mathrm{negint}}\right){,}{\mathrm{Not}}{}\left(\left({\mathrm{λ}}{-}{\mathrm{μ}}\right){::}{\mathrm{negint}}\right)\right)\right]$ (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 ${P}_{\mathrm{\lambda }}^{\mathrm{\mu }}\left(z\right)$ 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);
 $\frac{\sqrt{\frac{{2}{}{\mathrm{λ}}{+}{1}}{{\mathrm{π}}}}{}\sqrt{\left({\mathrm{λ}}{-}{\mathrm{μ}}\right){!}}}{\sqrt{\left({\mathrm{λ}}{+}{\mathrm{μ}}\right){!}}}$ (4)
 can be combined,
 > combine((4)) assuming posint;
 $\sqrt{\frac{\left({2}{}{\mathrm{λ}}{+}{1}\right){}\left({\mathrm{λ}}{-}{\mathrm{μ}}\right){!}}{{\mathrm{π}}{}\left({\mathrm{λ}}{+}{\mathrm{μ}}\right){!}}}$ (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];
 ${{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{{\mathrm{π}}}{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){}\stackrel{{&conjugate0;}}{{\mathrm{SphericalY}}{}\left({\mathrm{ρ}}{,}{\mathrm{ν}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right)}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{φ}}{=}{{\mathrm{δ}}}_{{\mathrm{λ}}{,}{\mathrm{ρ}}}{}{{\mathrm{δ}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}$ (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
 $\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{-}{\mathrm{θ}}{,}{\mathrm{φ}}\right){,}{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{-}{\mathrm{φ}}\right){}{{ⅇ}}^{{2}{}{I}{}{\mathrm{μ}}{}{\mathrm{φ}}}{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}\frac{{\mathrm{SphericalY}}{}\left({-}{1}{-}{\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){}\sqrt{{2}{}{\mathrm{λ}}{+}{1}}{}\sqrt{{\mathrm{Γ}}{}\left({-}{\mathrm{λ}}{+}{\mathrm{μ}}\right)}{}\sqrt{{\mathrm{Γ}}{}\left({\mathrm{λ}}{-}{\mathrm{μ}}{+}{1}\right)}}{\sqrt{{-}{1}{-}{2}{}{\mathrm{λ}}}{}\sqrt{{\mathrm{Γ}}{}\left({-}{\mathrm{λ}}{-}{\mathrm{μ}}\right)}{}\sqrt{{\mathrm{Γ}}{}\left({\mathrm{λ}}{+}{\mathrm{μ}}{+}{1}\right)}}{,}{\mathrm{And}}{}\left(\left({-}{\mathrm{λ}}{+}{\mathrm{μ}}\right){::}\left({\mathrm{Not}}{}\left({\mathrm{nonposint}}\right)\right){,}\left({-}{\mathrm{λ}}{-}{\mathrm{μ}}\right){::}\left({\mathrm{Not}}{}\left({\mathrm{nonposint}}\right)\right){,}\left({\mathrm{λ}}{+}{\mathrm{μ}}{+}{1}\right){::}\left({\mathrm{Not}}{}\left({\mathrm{nonposint}}\right)\right){,}\left({\mathrm{λ}}{-}{\mathrm{μ}}{+}{1}\right){::}\left({\mathrm{Not}}{}\left({\mathrm{nonposint}}\right)\right)\right)\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{-}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){}{{ⅇ}}^{{2}{}{I}{}{\mathrm{μ}}{}{\mathrm{φ}}}{,}{\mathrm{And}}{}\left({\mathrm{λ}}{::}{\mathrm{nonnegint}}{,}{\mathrm{μ}}{::}{\mathrm{integer}}{,}{\mathrm{μ}}{\le }{\mathrm{λ}}{,}{-}{\mathrm{λ}}{\le }{-}{\mathrm{μ}}\right)\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}{\left({-}{1}\right)}^{{\mathrm{μ}}}{}\stackrel{{&conjugate0;}}{{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{-}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right)}{,}{\mathrm{And}}{}\left({\mathrm{λ}}{::}{\mathrm{nonnegint}}{,}{\mathrm{μ}}{::}{\mathrm{integer}}{,}{\mathrm{μ}}{\le }{\mathrm{λ}}{,}{-}{\mathrm{λ}}{\le }{-}{\mathrm{μ}}\right)\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{2}{}{\mathrm{π}}{}{n}{+}{\mathrm{θ}}{,}{\mathrm{φ}}\right){,}{\mathrm{And}}{}\left({n}{::}{\mathrm{integer}}\right)\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}{+}\frac{{2}{}{\mathrm{π}}{}{n}}{{\mathrm{μ}}}\right){,}{\mathrm{And}}{}\left({n}{::}{\mathrm{integer}}{,}{\mathrm{μ}}{\ne }{0}\right)\right]\right]$ (7)

Examples

Expressing SphericalY in terms of LegendreP

 > $\mathrm{convert}\left(\mathrm{SphericalY}\left(\mathrm{λ},\mathrm{μ},\mathrm{θ},\mathrm{φ}\right),\mathrm{LegendreP}\right)$
 $\frac{{1}}{{2}}{}\frac{{\left({-}{1}\right)}^{{\mathrm{μ}}}{}\sqrt{\frac{{2}{}{\mathrm{λ}}{+}{1}}{{\mathrm{π}}}}{}\sqrt{\left({\mathrm{λ}}{-}{\mathrm{μ}}\right){!}}{}{{ⅇ}}^{{I}{}{\mathrm{φ}}{}{\mathrm{μ}}}{}{\mathrm{LegendreP}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{cos}}{}\left({\mathrm{θ}}\right)\right)}{\sqrt{\left({\mathrm{λ}}{+}{\mathrm{μ}}\right){!}}}$ (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

 > $\mathrm{convert}\left(\mathrm{SphericalY}\left(\mathrm{λ},\mathrm{μ},\mathrm{θ},\mathrm{φ}\right),\mathrm{LegendreP}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{λ}::\mathrm{posint},\mathrm{μ}::\mathrm{integer},\left|\mathrm{μ}\right|\le \mathrm{λ}$
 $\frac{{1}}{{2}}{}{\left({-}{1}\right)}^{{\mathrm{μ}}}{}\sqrt{\frac{\left({2}{}{\mathrm{λ}}{+}{1}\right){}\left({\mathrm{λ}}{-}{\mathrm{μ}}\right){!}}{{\mathrm{π}}{}\left({\mathrm{λ}}{+}{\mathrm{μ}}\right){!}}}{}{{ⅇ}}^{{I}{}{\mathrm{φ}}{}{\mathrm{μ}}}{}{\mathrm{LegendreP}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{cos}}{}\left({\mathrm{θ}}\right)\right)$ (9)

Special values

 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{SphericalY}\right)$
 $\left[\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}{0}{,}{\mathrm{And}}{}\left({2}{}{\mathrm{λ}}{+}{1}{=}{0}\right)\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}{0}{,}{\mathrm{And}}{}\left({\mathrm{μ}}{::}{\mathrm{integer}}{,}\left(\frac{{\mathrm{θ}}}{{\mathrm{π}}}\right){::}{\mathrm{even}}\right)\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}{0}{,}{\mathrm{And}}{}\left({\mathrm{ℜ}}{}\left({\mathrm{μ}}\right){<}{0}{,}\left(\frac{{\mathrm{θ}}}{{\mathrm{π}}}\right){::}{\mathrm{even}}\right)\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}{0}{,}{\mathrm{And}}{}\left({\mathrm{λ}}{::}{\mathrm{nonnegint}}{,}{\mathrm{μ}}{::}{\mathrm{posint}}{,}{\mathrm{λ}}{<}\left|{\mathrm{μ}}\right|\right)\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}{0}{,}{\mathrm{And}}{}\left({\mathrm{λ}}{::}{\mathrm{nonnegint}}{,}\left(\frac{{\mathrm{θ}}}{{\mathrm{π}}}\right){::}{\mathrm{integer}}{,}{\mathrm{μ}}{::}{\mathrm{posint}}\right)\right]{,}\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}\frac{{1}}{{2}}{}{\left({-}{1}\right)}^{\frac{{\mathrm{λ}}{}\left({2}{}{\mathrm{floor}}{}\left(\frac{{1}}{{2}}{}\frac{{\mathrm{θ}}}{{\mathrm{π}}}\right){}{\mathrm{π}}{-}{\mathrm{θ}}\right)}{{\mathrm{π}}}}{}\sqrt{\frac{{2}{}{\mathrm{λ}}{+}{1}}{{\mathrm{π}}}}{,}{\mathrm{And}}{}\left({\mathrm{λ}}{::}{\mathrm{nonnegint}}{,}\left(\frac{{\mathrm{θ}}}{{\mathrm{π}}}\right){::}{\mathrm{integer}}{,}{\mathrm{μ}}{=}{0}\right)\right]\right]$ (10)

Hypergeometric representation

 > $\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{SphericalY},\mathrm{hypergeom}\right)$
 $\left[{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}\frac{{1}}{{2}}{}\frac{{\left({-}{1}\right)}^{{\mathrm{μ}}}{}\sqrt{\frac{{2}{}{\mathrm{λ}}{+}{1}}{{\mathrm{π}}}}{}\sqrt{\left({\mathrm{λ}}{-}{\mathrm{μ}}\right){!}}{}{{ⅇ}}^{{I}{}{\mathrm{φ}}{}{\mathrm{μ}}}{}{\left({\mathrm{cos}}{}\left({\mathrm{θ}}\right){+}{1}\right)}^{\frac{{1}}{{2}}{}{\mathrm{μ}}}{}{\mathrm{hypergeom}}{}\left(\left[{-}{\mathrm{λ}}{,}{\mathrm{λ}}{+}{1}\right]{,}\left[{1}{-}{\mathrm{μ}}\right]{,}\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{\mathrm{cos}}{}\left({\mathrm{θ}}\right)\right)}{\sqrt{\left({\mathrm{λ}}{+}{\mathrm{μ}}\right){!}}{}{\left({\mathrm{cos}}{}\left({\mathrm{θ}}\right){-}{1}\right)}^{\frac{{1}}{{2}}{}{\mathrm{μ}}}{}{\mathrm{Γ}}{}\left({1}{-}{\mathrm{μ}}\right)}{,}{\mathrm{And}}{}\left({\mathrm{Not}}{}\left(\left({\mathrm{λ}}{+}{\mathrm{μ}}\right){::}{\mathrm{negint}}\right){,}{\mathrm{Not}}{}\left(\left({\mathrm{λ}}{-}{\mathrm{μ}}\right){::}{\mathrm{negint}}\right){,}{\mathrm{Not}}{}\left(\left({1}{-}{\mathrm{μ}}\right){::}{\mathrm{nonposint}}\right)\right)\right]$ (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.