# Spherical coordinates

January 29, 2021 — September 23, 2024

The **spherical harmonic transform** is the natural extension of the Fourier series to the surface of a sphere, used for functions defined on spherical domains. It expresses a function in terms of spherical harmonics, \(Y_{\ell m}(\theta, \phi)\), where \(\ell\) and \(m\) are integers that index the degree and order of the harmonics, respectively. The spherical harmonics are the angular portion of the solution to Laplace’s equation in spherical coordinates, making this transform useful in solving problems on spherical domains, like those in geophysics or astrophysics.

cf the Hankel transform, which is kind of a radial version of the Fourier transform, characterising dependence upon radius. That is typically used for problems with radial symmetry, such as wave propagation, heat flow, or Laplace’s equation in circular geometries. The Hankel transform of order \(n\) is defined using the Bessel function of the first kind, \(J_n(x)\), as the kernel: \[ \mathcal{H}_n[f(r)](k) = \int_0^\infty f(r) J_n(kr) r \, dr \]

Spherical harmonics handle the *angular* dependence of functions on a sphere, while the Hankel Transform handles the *radial* dependence. When solving PDEs in spherical coordinates, it’s common to represent the angular part of the solution using spherical harmonics and the radial part using the Hankel.

## 1 Tools

## 2 References

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*Geochemistry, Geophysics, Geosystems*.

*Advances in Computational Mathematics*.

*Journal of Computational and Applied Mathematics*, Numerical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials,.

*Advances in Computational Mathematics*.