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In of R''n'') which satisfies Laplace's Equation , i.e.

:
rac{\partial^2f}{\partial x_1^2} +
rac{\partial^2f}{\partial x_2^2} +
\cdots +
rac{\partial^2f}{\partial x_n^2} = 0

everywhere 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 .

A function that satisfies \Delta f \ge 0 is said to be '' Subharmonic ''.


EXAMPLES

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 three variables are:


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.



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.

If ''f'' is a harmonic function on ''U'', then all Partial Derivative s of ''f'' are also harmonic functions on ''U''.

In several ways, the harmonic functions are real analogues to Holomorphic Function s. All harmonic functions are Analytic , i.e. they can be locally expressed as Power Series . This is a general fact about Elliptic Operator s, of which the Laplacian is a major example.


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 Operator s. 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 Angle s.


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 '' 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 .


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) = rac{1}{\omega_n r^{n-1}}\oint_{\partial B(x,r)} u \, dS
= rac{n}{\omega_n r^n}\int_{B (x,r)} u \, dV


where \omega_n is the surface area of the Unit Sphere 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 ).


GENERAL THEORY

The generalization of the study of harmonic functions is the study of Harmonic Form s on Riemannian Manifold s, and is known as Cohomology .


SEE ALSO