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based on an earlier version of the Wolfram Language.
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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.
Evaluate for complex arguments and orders:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
SphericalHarmonicY threads element-wise over lists:
TraditionalForm formatting:
SphericalHarmonicY can be applied to a power series:
SphericalHarmonicY is an eigenfunction of the spherical part of the Laplace operator:
Use FunctionExpand to expand SphericalHarmonicY for half-integers and :
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