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Examples of harmonic functions of two variables are: The real or imaginary part of any holomorphic function.; The function (,) = ; this is a special case of the example above, as (,) = (+), and + is a holomorphic function.
Historically, harmonic functions first referred to the solutions of Laplace's equation. [2] This terminology was extended to other special functions that solved related equations, [3] then to eigenfunctions of general elliptic operators, [4] and nowadays harmonic functions are considered as a generalization of periodic functions [5] in function spaces defined on manifolds, for example as ...
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). For example, for any the formula (,,) = (+) defines a homogeneous polynomial of degree with domain and codomain , which happens to be independent of . This polynomial is ...
Weakly harmonic function; Weyl's lemma (Laplace equation) This page was last edited on 22 November 2022, at 22:46 (UTC). Text is available under the Creative ...
Formally, the definition can be stated as follows. Let be a subset of the Euclidean space and let : {} be an upper semi-continuous function.Then, is called subharmonic if for any closed ball (,) ¯ of center and radius contained in and every real-valued continuous function on (,) ¯ that is harmonic in (,) and satisfies () for all on the boundary (,) of (,), we have () for all (,).
Conjugate harmonic functions (and the transform between them) are also one of the simplest examples of a Bäcklund transform (two PDEs and a transform relating their solutions), in this case linear; more complex transforms are of interest in solitons and integrable systems.
In mathematics and mathematical physics, potential theory is the study of harmonic functions.. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which ...
Mechanical examples include pendulums (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a ...