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In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplace's equation, i.e. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ...
In the mathematics of probability, a stochastic process is a random function. ...
In mathematics, a derivative is the rate of change of a quantity. ...
Partial plot of a function f. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In mathematics, Laplaces equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. ...
everywhere on U. This is also often written as - or
There also exists a seemingly weaker definition that is equivalent. Indeed a function is harmonic if and only if it is weakly harmonic. In mathematics, a function is weakly harmonic in a domain D if for all with compact support in D and continuous second derivatives, where Δ is the Laplacian. ...
A function that satisfies is said to be subharmonic. In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations and complex analysis. ...
Examples
Examples of harmonic functions of two variables are: - the real and imaginary part of any holomorphic function
- the function
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- f(x1, x2) = ln(x12 + x22)
- defined on R2 {0} (e.g. the electric potential due to a line charge, and the gravity potential due to a long cylindrical mass)
- the function f(x1, x2) = exp(x1)sin(x2).
Examples of harmonic functions of three variables are: Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ...
The exponential function is one of the most important functions in mathematics. ...
Examples of harmonic functions of n variables are: Electric potential is the potential energy per unit of charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. ...
Gravity is a force of attraction that acts between bodies that have mass. ...
- the constant, linear and affine functions on all of Rn (for example, the electric potential between the plates of a capacitor, and the gravity potential of a slab)
- the function f(x1,...,xn) = (x12 + ... + xn2)1 −n/2 on Rn {0} for n ≥ 2.
Capacitors: SMD ceramic at top left; SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at bottom right. ...
Remarks The set of harmonic functions on a given open set U can be seen as the kernel of the Laplace operator Δ and is therefore a vector space over R: sums, differences and scalar multiples of harmonic functions are again harmonic. In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
If f is a harmonic function on U, then all partial derivatives of f are also harmonic functions on U. In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...
In several ways, the harmonic functions are real analogues to holomorphic functions. All harmonic functions are analytic, i.e. they can be locally expressed as power series. This is a general fact about elliptic operators, of which the Laplacian is a major example. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
In mathematics, an analytic function is a function that is locally given by a convergent power series. ...
In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...
In mathematics, an elliptic operator is one of the major types of differential operator P. It can also be defined on spaces of complex-valued functions, or some more general function-like objects. ...
Connections with complex function theory The real and imaginary part of any holomorphic function yield harmonic functions on R2. Conversely there is an operator taking a harmonic function u on a region in R2 to its harmonic conjugate v, for which u+iv is a holomorphic function; here v is well-defined up to a real constant. This is well known in applications as (essentially) the Hilbert transform; it is also a basic example in mathematical analysis, in connection with singular integral operators. Geometrically u and v are related as having orthogonal trajectories, away from the zeroes of the underlying holomorphic function; the contours on which u and v are constant cross at right angles. In this regard, u+iv would be the complex potential, where u is the potential function and v is the stream function. In mathematics, the harmonic conjugate of a harmonic real-valued function of two variables u(x,y), is a function v(x,y) such that v is harmonic and u and v satisfy the Cauchy-Riemann equations, that is, the complex-valued function u(x,y)+iv(x,y) = f...
In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ...
The Hilbert transform, in red, of a square wave, in blue In mathematics and in signal processing, the Hilbert transform, here denoted , of a real-valued function, , is obtained by convolving signal with to obtain . ...
Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ...
This article is about angles in geometry. ...
The term potential function can mean more than one thing. ...
In fluid dynamics, the stream function is defined for two-dimensional flows. ...
Properties of harmonic functions Some important properties of harmonic functions can be deduced from Laplace's equation.
The maximum principle Harmonic functions satisfy the following maximum principle: if K is any compact subset of U, then f, restricted to K, attains its maximum and minimum on the boundary of K. If U is connected, this means that f cannot have local maxima or minima, other than in ccse of the exceptional case where f is constant. In mathematics, the maximum principle in harmonic analysis states that if f is a harmonic function, then f cannot exhibit a true local maximum within the domain of definition of f. ...
Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly...
A graph illustrating local min/max and global min/max points In mathematics, maxima and minima, also known as extrema, are points in the domain of a function at which the function takes the largest (maximum), or smallest (minimum) value either within a given neighbourhood (local extrema), or on the...
In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of...
Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ...
In mathematics a constant function is a function whose values do not vary and thus are constant. ...
The mean value property If B(x,r) is a ball with center x and radius r which is completely contained in U, then the value f(x) of the harmonic function f at the center of the ball is given by the average value of f on the surface of the ball; this average value is also equal to the average value of f in the interior of the ball. In other words
where ωn is the surface area of the unit sphere in n dimensions. In mathematics, unit ball and unit sphere refer to a ball with radius equal to 1. ...
Liouville's theorem If f is a harmonic function defined on all of Rn which is bounded above or bounded below, then f is constant (compare Liouville's theorem for functions of a complex variable). Liouvilles theorem has various meanings: In complex analysis, see Liouvilles theorem (complex analysis). ...
Generalizations One generalization of the study of harmonic functions is the study of harmonic forms on Riemannian manifolds, and it is related to the study of cohomology. Also, it is possible to define harmonic vector-valued functions, or harmonic maps of two Riemannian manifolds, which are critical points of a generalized Dirichlet energy functional (this includes harmonic functions as a special case, a result known as Dirichlet principle). These kind of harmonic maps appear in the theory of minimal surfaces. For example, a curve, that is, a map from an interval in R to a Riemannian manifold, is a harmonic map if and only if it is a geodesic. In mathematics, Hodge theory is the study of the consequences for the algebraic topology of a smooth manifold M of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M. It was developed by W. V. D. Hodge in the 1930s as an extension...
In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ...
In mathematics, Dirichlets principle in potential theory states that the harmonic function on a domain with boundary condition on can be obtained as the minimizer of the Dirichlet integral amongst all functions such that on , provided only that there exists one such function making the Dirichlet integral finite. ...
In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ...
External links - Eric W. Weisstein, Harmonic Function at MathWorld.
- Harmonic Functions Module by John H. Mathews
MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...
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