Read More
Date: 24-5-2018
636
Date: 30-5-2018
944
Date: 12-7-2018
590
|
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
(11) |
|||
(12) |
Other solutions may be obtained by differentiation, such as
(13) |
|||
(14) |
(15) |
|||
(16) |
and
(17) |
Harmonic functions containing azimuthal dependence include
(18) |
|||
(19) |
The Poisson kernel
(20) |
is another harmonic function.
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.
|
|
دور في الحماية من السرطان.. يجب تناول لبن الزبادي يوميا
|
|
|
|
|
العلماء الروس يطورون مسيرة لمراقبة حرائق الغابات
|
|
|
|
|
ضمن أسبوع الإرشاد النفسي.. جامعة العميد تُقيم أنشطةً ثقافية وتطويرية لطلبتها
|
|
|