In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, spherical harmonics are the angular portion of a set of solutions to Laplace's equation

Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

. Represented in a system of spherical coordinates

Spherical coordinate system

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith direction, and the azimuth angle of...

, Laplace's spherical harmonics are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre Simon de Laplace in 1782. Spherical harmonics are important in many theoretical and practical applications, particularly in the computation of atomic orbital

Atomic orbital

An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

 electron configuration

Electron configuration

In atomic physics and quantum chemistry, electron configuration is the arrangement of electrons of an atom, a molecule, or other physical structure...

s, representation of gravitational field

Gravitational field

The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

s, geoid

Geoid

The geoid is that equipotential surface which would coincide exactly with the mean ocean surface of the Earth, if the oceans were in equilibrium, at rest , and extended through the continents . According to C.F...

s, and the magnetic field

Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

s of planetary bodies

Planet

A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

 and stars, and characterization of the cosmic microwave background radiation

Cosmic microwave background radiation

In cosmology, cosmic microwave background radiation is thermal radiation filling the observable universe almost uniformly....

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