Any real function 
with continuous second partial
derivatives which satisfies Laplace's equation,
 |
(1) |
is called a harmonic function. Harmonic functions are called potential functions in physics and engineering. Potential functions are extremely useful, for example, in electromagnetism, where they reduce the study of a 3-component vector field to a 1-component scalar function. A scalar harmonic function is called a scalar potential, and a vector harmonic function is called a vector potential.
To find a class of such functions in the plane, write the Laplace's equation in polar coordinates
 |
(2) |
and consider only radial solutions
 |
(3) |
This is integrable by quadrature, so define 
,
 |
(4) |
 |
(5) |
 |
(6) |
 |
(7) |
 |
(8) |
 |
(9) |
so the solution is
 |
(10) |
Ignoring the trivial additive and multiplicative constants, the general pure radial solution then becomes
Other solutions may be obtained by differentiation, such as
and
 |
(17) |
Harmonic functions containing azimuthal dependence include
The Poisson kernel
 |
(20) |
is another harmonic function.
See also
Conformal Mapping, Dirichlet Problem, Harmonic Analysis, Harmonic Decomposition, Harnack's Inequality, Harnack's Principle, Kelvin Transformation, Laplace's Equation, Poisson Integral, Poisson Kernel, Scalar Potential, Schwarz Reflection Principle, Subharmonic Function, Vector Potential
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References
Ash, J. M. (Ed.). Studies in Harmonic Analysis. Washington, DC: Math. Assoc. Amer., 1976.Axler, S.; Bourdon, P.; and Ramey, W. Harmonic Function Theory. Springer-Verlag, 1992.Benedetto, J. J. Harmonic Analysis and Applications. Boca Raton, FL: CRC Press, 1996.Cohn, H. Conformal Mapping on Riemann Surfaces. New York: Dover, 1980.Krantz, S. G. "Harmonic Functions." §1.4.1 and Ch. 7 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 16 and 89-101, 1999.Weisstein, E. W. "Books about Potential Theory." http://www.ericweisstein.com/encyclopedias/books/PotentialTheory.html.
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Cite this as:
Weisstein, Eric W. "Harmonic Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HarmonicFunction.html