I learned about Spherical Harmonics in a Topics in Nonparametric Statistics course. Spherical harmonics can be used to test for multivariate normality. Spherical Harmonics are a set of orthogonal functions defined on the surface of a sphere. They are used in various fields of science and engineering to represent complex functions over spherical domains. Mathematically, Spherical Harmonics are solutions to the angular part of Laplace’s equation in spherical coordinates.

Laplace’s equation is given by:

$$

\nabla^2 \phi = 0

$$

In Cartesian coordinates, if \((\phi)\) is a function of (x), (y), and (z), the Laplacian operator is:

$$

\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}

$$

## Mathematical Definition

The Spherical Harmonics \( Y_{l}^{m}(\theta, \phi) \) are given by: \[ Y_{l}^{m}(\theta, \phi) = \sqrt{\frac{(2l+1)}{4\pi} \frac{(l-m)!}{(l+m)!}} \cdot P_{l}^{m}(\cos \theta) \cdot e^{im\phi} \] where:

– \( l \) is the degree of the harmonic,

– \( m \) is the order of the harmonic,

– \( \theta \) is the polar angle,

– \( \phi \) is the azimuthal angle,

– \( P_{l}^{m} \) are the associated Legendre polynomials.

## Applications of Spherical Harmonics

Spherical Harmonics are used in various fields, including:

– **Quantum Mechanics:** To describe the angular part of wave functions of particles in spherical potentials (e.g., electrons in atoms).

– **Geophysics and Astronomy:** For modeling gravitational and magnetic fields, and planetary surfaces.

– **Computer Graphics:** In environment mapping and image-based lighting.

– **Acoustics: **For sound field analysis and room acoustics modeling.

We can visualize Spherical Harmonics in a 3D plot. In this plot, the plot we represent a specific Spherical Harmonic function \(Y_3^2\) with \(l=3\) and \(m=2\).