Are spherical harmonics orthogonal?
They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics.
What do spherical harmonics describe?
Spherical harmonics are defined as the eigenfunctions of the angular part of the Laplacian in three dimensions. As a result, they are extremely convenient in representing solutions to partial differential equations in which the Laplacian appears.
What is the orthonormal basis of spherical harmonics?
Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics.
What is spherical harmonic polynomial representation?
Harmonic polynomial representation. For any , the space of spherical harmonics of degree is just the space of restrictions to the sphere of the elements of . As suggested in the introduction, this perspective is presumably the origin of the term “spherical harmonic” (i.e., the restriction to the sphere of a harmonic function).
What is the parity of spherical harmonics?
The spherical harmonics have definite parity. That is, they are either even or odd with respect to inversion about the origin. Inversion is represented by the operator . Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with
What are the Legendre functions and the spherical harmonics?
Here we build on these and introduce the associated Legendre functions Pmℓ (x) in the first part of the chapter, and the spherical harmonics Ymℓ (θ, ϕ) in the second part. Spherical harmonics are used extremely widely in physics.