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.


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Basic Examples  (5)

Evaluate symbolically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (28)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

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]:

Visualization  (3)

Plot the SphericalHarmonicY function for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot the absolute value of the SphericalHarmonicY function in three dimensions:

Function Properties  (7)

Domain of SphericalHarmonicY:

Domain for complex values:

The range for SphericalHarmonicY:

The range for complex values:

SphericalHarmonicY is an even function with respect to θ and ϕ for even-order m:

SphericalHarmonicY is an odd function with respect to θ and ϕ for odd-order m:

SphericalHarmonicY is a periodic function with respect to θ and ϕ:

SphericalHarmonicY has the mirror property Y_1^2(pi/6,TemplateBox[{z}, Conjugate])=TemplateBox[{{{Y, _, 1, ^, 2}, (, {{pi, /, 6}, ,, z}, )}}, Conjugate]:

SphericalHarmonicY threads elementwise over lists:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to ϕ:

First derivative with respect to θ:

Higher derivatives with respect to θ:

Plot the absolute values of the higher derivatives of with respect to :

Formula for the ^(th) derivative with respect to :

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Series Expansions  (4)

Find the Taylor expansion using Series:

General term in the series expansion using SeriesCoefficient:


Taylor expansion at a generic point:

Generalizations & Extensions  (1)

SphericalHarmonicY can be applied to a power series:

Applications  (1)

SphericalHarmonicY is an eigenfunction of the spherical part of the Laplace operator:

Properties & Relations  (1)

Use FunctionExpand to expand SphericalHarmonicY[n,m,θ,ϕ] for half-integers and :

Introduced in 1988