Any real-valued function
with continuous second Partial Derivatives which satisfies
Laplace's Equation
![\begin{displaymath}
\nabla^2 u(x,y)=0
\end{displaymath}](h_449.gif) |
(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
![\begin{displaymath}
u_{rr}+{1\over r} u_r+{1\over r^2} u_{\theta\theta}=0,
\end{displaymath}](h_450.gif) |
(2) |
and consider only radial solutions
![\begin{displaymath}
u_{rr}+{1\over r} u_r=0.
\end{displaymath}](h_451.gif) |
(3) |
This is integrable by quadrature, so define
,
![\begin{displaymath}
{dv\over dr}+{1\over r}v=0
\end{displaymath}](h_453.gif) |
(4) |
![\begin{displaymath}
{dv\over v}=-{dr\over r}
\end{displaymath}](h_454.gif) |
(5) |
![\begin{displaymath}
\ln\left({v\over A}\right)=-\ln r
\end{displaymath}](h_455.gif) |
(6) |
![\begin{displaymath}
{v\over A}={1\over r}
\end{displaymath}](h_456.gif) |
(7) |
![\begin{displaymath}
v={du\over dr}={A\over r}
\end{displaymath}](h_457.gif) |
(8) |
![\begin{displaymath}
du = A{dr\over r},
\end{displaymath}](h_458.gif) |
(9) |
so the solution is
![\begin{displaymath}
u=A\ln r.
\end{displaymath}](h_459.gif) |
(10) |
Ignoring the trivial additive and multiplicative constants, the general pure radial solution then becomes
![\begin{displaymath}
u=\ln[(x-a)^2+(y-b)^2]^{1/2} = {\textstyle{1\over 2}}\ln\left[{(x-a)^2+(y-b)^2}\right].
\end{displaymath}](h_460.gif) |
(11) |
Other solutions may be obtained by differentiation, such as
and
![\begin{displaymath}
\tan^{-1}\left({y-b\over x-a}\right).
\end{displaymath}](h_465.gif) |
(16) |
Harmonic functions containing azimuthal dependence include
The Poisson Kernel
![\begin{displaymath}
u(r,R,\theta,\phi)={R^2-r^2\over R^2-2rR\cos(\theta-\phi)+r^2}
\end{displaymath}](h_468.gif) |
(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.
© 1996-9 Eric W. Weisstein
1999-05-25