scientific fields. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal...

This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree ℓ=10{\displaystyle \ell =10}. Some of these...

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH...

zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions...

harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, the spin-weighted harmonics are U(1) gauge fields rather than scalar fields: mathematically...

differential equations) in spherical coordinates. Associated Legendre polynomials play a vital role in the definition of spherical harmonics. These functions are...

fit the measurements to a set of spherical harmonics. This was first done by Carl Friedrich Gauss. Spherical harmonics are functions that oscillate over...

infinity, making A = 0. This does not affect the angular portion of the spherical harmonics. Stewart, James. Calculus : Early Transcendentals. 7th ed., Brooks/Cole...

transformation of complex spherical harmonics to real form is by a unitary transformation, we can simply substitute real irregular solid harmonics and real multipole...

In quantum mechanics, the spinor spherical harmonics (also known as spin spherical harmonics, spinor harmonics and Pauli spinors) are special functions...

needs. Furthermore, there was no widely accepted formulation of spherical harmonics for acoustics, so one was borrowed from chemistry, quantum mechanics...

group can be defined simply in terms of integrals of products of spherical harmonics and their complex conjugates. The addition of spins in quantum-mechanical...

class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates...

Earth's mantle. Spherical harmonics are often used to approximate the shape of the geoid. The current best such set of spherical harmonic coefficients is...

angular portions of the solutions to such equations take the form of spherical harmonics. Another application is ergonomic design, where r is the arm length...

important role here via the connection to the angular dependence of the spherical harmonics for the different orbitals around each atom. The term "azimuthal...

{k} }})Y_{\ell }^{m*}({\hat {\mathbf {r} }}),} where Yℓm are the spherical harmonics and the superscript * denotes complex conjugation. Note that the...

3-dimensional spherical harmonics. These shapes are not unique, and any linear combination is valid, like a transformation to cubic harmonics, in fact it...

corresponding radiation pattern of the angular part of vector spherical harmonics. The harmonics N o e m 1 {\displaystyle \mathbf {N} _{^{e}_{o}m1}} correspond...

transform into its radial and spherical components leads to topics such as Bessel functions and spherical harmonics. Harmonic analysis on tube domains is...

m\leq \ell \right\},} where Y m ℓ {\displaystyle Y_{m}^{\ell }} are spherical harmonics. Its elements are square integrable complex-valued functions on the...

angular or spherical part. The solution to the spherical part of the equation can be expressed as a series of Laplace's spherical harmonics, simplifying...

"Spherical Harmonic". Powells.com. Retrieved 15 December 2009. Text with sinusoidal prose poems Asaro describes the mathematics of spherical harmonics...

often partially replaced by cubic harmonics for a number of reasons. These harmonics are usually named tesseral harmonics in the field of condensed matter...

remain. The 3-jm symbols give the integral of the products of three spherical harmonics ∫ Y l 1 m 1 ( θ , φ ) Y l 2 m 2 ( θ , φ ) Y l 3 m 3 ( θ , φ ) sin...

1st harmonic; the other harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also...

the years models consisting of an increasing but limited number of spherical harmonics have been produced. Maps produced have included free-air gravity...

D-matrix elements with second index equal to zero are proportional to spherical harmonics and associated Legendre polynomials, normalized to unity and with...

of the harmonics on the unit n-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and over time "harmonic" was used...

In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions...