Harmonic function

Harmonic function

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 R"n") which satisfies Laplace's equation, i.e.

:frac{partial^2f}{partial x_1^2} +frac{partial^2f}{partial x_2^2} +cdots +frac{partial^2f}{partial x_n^2} = 0everywhere on "U". This is also often written as: abla^2 f = 0 or Delta f = 0.

There also exists a seemingly weaker definition that is equivalent. Indeed a function is harmonic if and only if it is weakly harmonic.

Harmonic functions can be defined on an arbitrary Riemannian manifold, using the Laplace-de Rham operatorDelta. In this context, a function is called "harmonic" if Delta f = 0.

A C^2 function that satisfies Delta f ge 0 is said to be "subharmonic".


Examples of harmonic functions of two variables are:
* the real and imaginary part of any holomorphic function
* the function ::"f"("x"1, "x"2) = ln("x"12 + "x"22): 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"("x"1, "x"2) = exp("x"1)sin("x"2).

Examples of harmonic functions of "n" variables are:

*the constant, linear and affine functions on all of R"n" (for example, the electric potential between the plates of a capacitor, and the gravity potential of a slab)
*the function "f"("x"1,...,"x""n") = ("x"12 + ... + "x""n"2)1 −"n"/2 on R"n" {0} for "n" ≥ 2.

Examples of harmonic functions of three variables are given in the table below with r^2=x^2+y^2+z^2. Harmonic functions are determined by their singularities. The singular points of the harmonic functions below are expressed as "charges" and "charge densities" using the terminology of electrostatics, and so the corresponding harmonic function will be proportional to the electrostatic potential due to these charge distributions. Each function below will yield another harmonic function when multiplied by a constant, rotated, and/or has a constant added. The inversion of each function will yield another harmonic function which has singularities which are the images of the original singularities in a spherical "mirror". Also, the sum of any two harmonic functions will yield another harmonic function.



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.

If "f" is a harmonic function on "U", then all partial derivatives of "f" are also harmonic functions on "U". The Laplace operator Δ and the partial derivative operator will commute on this class of functions.

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.

The uniform limit of a convergent sequence of harmonic functions is still harmonic. This is true because any continuous function satisfying the mean value property is harmonic. Consider the sequence on (-infty, 0)× R defined by scriptstyle f_n(x,y) = frac1n exp(nx)cos(ny). This sequence is harmonic and converges uniformly to the zero function; however note that the partial derivatives are not uniformly convergent to the zero function (the derivative of the zero function). This example shows the importance on relying on the mean value property and continuity to argue the limit is harmonic.

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.

Properties of harmonic functions

Some important properties of harmonic functions can be deduced from Laplace's equation.

The regularity theorem for harmonic functions

Harmonic functions are infinitely differentiable. In fact, harmonic functions are real analytic.

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 the exceptional case where "f" is constant. Similar properties can be shown for subharmonic functions.

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

u(x) = frac{1}{nomega_n r^{n-1int_{partial B(x,r)} u , dS = frac{1}{omega_n r^n}int_{B (x,r)} u , dV

where omega_n is the volume of the unit ball in "n" dimensions.

Liouville's theorem

If "f" is a harmonic function defined on all of R"n" which is bounded above or bounded below, then "f" is constant (compare Liouville's theorem for functions of a complex variable).


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.

ee also

*Dirichlet problem
*Heat equation
*Laplace's equation
*Poisson's equation
*Quadrature domains
*Subharmonic function


* L.C. Evans, 1998. "Partial Differential Equations". American Mathematical Society.
*D. Gilbarg, N. Trudinger "Elliptic Partial Differential Equations of Second Order". ISBN 3-540-41160-7.
* Q. Han, F. Lin, 2000, "Elliptic Partial Differential Equations", American Mathematical Society

External links

* [http://math.fullerton.edu/mathews/c2003/HarmonicFunctionMod.html Harmonic Functions Module by John H. Mathews]
* [http://www.axler.net/HFT.html Harmonic Function Theory by S.Axler, Paul Bourdon, and Wade Ramey]

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • harmonic function — ▪ mathematics       mathematical function of two variables having the property that its value at any point is equal to the average of its values along any circle around that point, provided the function is defined within the circle. An infinite… …   Universalium

  • harmonic function — harmoninė funkcija statusas T sritis automatika atitikmenys: angl. harmonic function vok. harmonische Funktion, f rus. гармоническая функция, f pranc. fonction harmonique, f …   Automatikos terminų žodynas

  • harmonic function — harmoninė funkcija statusas T sritis fizika atitikmenys: angl. harmonic function vok. harmonische Funktion, f rus. гармоническая функция, f pranc. fonction harmonique, f …   Fizikos terminų žodynas

  • harmonic function — noun A function of two, three or n variables which is a solution to any of Laplaces equations …   Wiktionary

  • Weakly harmonic function — In mathematics, a function f is weakly harmonic in a domain D if :int D f, Delta g = 0 for all g with compact support in D and continuous second derivatives, where Delta; is the Laplacian. This definition is weaker than the definition of harmonic …   Wikipedia

  • Harmonic conjugate — For geometric conjugate points, see Projective harmonic conjugates. In mathematics, a function defined on some open domain is said to have as a conjugate a function if and only if they are respectively real and imaginary part of a holomorphic… …   Wikipedia

  • Harmonic measure — In mathematics, harmonic measure is a concept that arises in the theory of harmonic functions, where it can be used to estimate the modulus of an analytic function inside a domain D given bounds on the modulus on the boundary of the domain. In a… …   Wikipedia

  • Harmonic map — A (smooth) map φ: M → N between Riemannian manifolds M and N is called harmonic if it is a critical point of the energy functional E (φ).This functional E will be defined precisely below mdash;one way of understanding it is to imagine that M is… …   Wikipedia

  • Harmonic (mathematics) — In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving… …   Wikipedia

  • Harmonic — Har*mon ic (h[aum]r*m[o^]n [i^]k), Harmonical Har*mon ic*al ( [i^]*kal), a. [L. harmonicus, Gr. armoniko s; cf. F. harmonique. See {Harmony}.] 1. Concordant; musical; consonant; as, harmonic sounds. [1913 Webster] Harmonic twang! of leather, horn …   The Collaborative International Dictionary of English

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.