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单词 Harmonic Function
释义

Harmonic Function

Any real-valued function with continuous second Partial Derivatives which satisfiesLaplace's Equation

(1)

is called a harmonic function. Harmonic functions are called Potential Functions in physicsand engineering. Potential functions are extremely useful, for example, inelectromagnetism, where they reduce the study of a 3-component Vector Field toa 1-component Scalar Function. A scalar harmonic function is called a Scalar Potential, and a vectorharmonic 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
(11)


Other solutions may be obtained by differentiation, such as

(12)
(13)


(14)
(15)

and
(16)

Harmonic functions containing azimuthal dependence include
(17)
(18)

The Poisson Kernel
(19)

is another harmonic function.

See also Scalar Potential, Vector Potential


References

Potential Theory

Ash, J. M. (Ed.) Studies in Harmonic Analysis. Washington, DC: Math. Assoc. Amer., 1976.

Axler, S.; Pourdon, 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.
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