# SphericalHarmonicY

SphericalHarmonicY[l,m,θ,ϕ]

gives the spherical harmonic .

# Details • 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.

# Examples

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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(34)

### 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(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:

The real range of :

The range for complex values: 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: 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 not injective: is not surjective: 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:

### 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  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(2)

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

Plot hydrogen orbital densities for quantum numbers , , :

Plot :

## Properties & Relations(2)

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

Re-express spherical harmonics in Cartesian coordinates: