gives the spherical harmonic .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The spherical harmonics are orthonormal with respect to integration over the surface of the unit sphere.
- For , where is the associated Legendre function.
- For , .
- For certain special arguments, SphericalHarmonicY automatically evaluates to exact values.
- SphericalHarmonicY can be evaluated to arbitrary numerical precision.
- SphericalHarmonicY automatically threads over lists.
Examplesopen allclose all
Basic Examples (5)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Numerical Evaluation (4)
Specific Values (4)
Evaluate SphericalHarmonicY symbolically for integer orders:
Evaluate SphericalHarmonicY symbolically for noninteger orders:
Evaluate SphericalHarmonicY symbolically for :
SphericalHarmonicY for symbolic l and m:
Find the first positive maximum of SphericalHarmonicY[2,2,θ,Pi/2]:
Plot the SphericalHarmonicY function for various orders:
Plot the absolute value of the SphericalHarmonicY function in three dimensions:
Function Properties (13)
For integer and , is defined for all complex and :
For , it is defined as a real function for all real and :
For other values of , it is typically not defined as a real function:
is an even function with respect to for even-order :
It is an odd function with respect to for odd-order :
SphericalHarmonicY is a periodic function with respect to θ and ϕ:
SphericalHarmonicY threads elementwise over lists:
is an analytic function of and for integer and :
For , it is analytic over the reals:
is neither non-decreasing nor non-increasing as a function of :
is neither non-positive nor non-negative:
does not have either a singularity or a discontinuity over the complexes for integer and :
For , it is nonsingular over the reals as well:
is neither convex nor concave:
Compute the indefinite integral using Integrate:
Series Expansions (4)
Find the Taylor expansion using Series:
General term in the series expansion using SeriesCoefficient:
Generalizations & Extensions (1)
SphericalHarmonicY can be applied to a power series:
SphericalHarmonicY is an eigenfunction of the spherical part of the Laplace operator:
Plot hydrogen orbital densities for quantum numbers , , :
Properties & Relations (2)
Use FunctionExpand to expand SphericalHarmonicY[n,m,θ,ϕ] for half-integers and :
Wolfram Research (1988), SphericalHarmonicY, Wolfram Language function, https://reference.wolfram.com/language/ref/SphericalHarmonicY.html.
Wolfram Language. 1988. "SphericalHarmonicY." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SphericalHarmonicY.html.
Wolfram Language. (1988). SphericalHarmonicY. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SphericalHarmonicY.html