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Harmonic Function

Any real-valued function $u(x,y)$ with continuous second Partial Derivatives which satisfies Laplace's Equation

\nabla^2 u(x,y)=0
\end{displaymath} (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

u_{rr}+{1\over r} u_r+{1\over r^2} u_{\theta\theta}=0,
\end{displaymath} (2)

and consider only radial solutions
u_{rr}+{1\over r} u_r=0.
\end{displaymath} (3)

This is integrable by quadrature, so define $v\equiv du/dr$,
{dv\over dr}+{1\over r}v=0
\end{displaymath} (4)

{dv\over v}=-{dr\over r}
\end{displaymath} (5)

\ln\left({v\over A}\right)=-\ln r
\end{displaymath} (6)

{v\over A}={1\over r}
\end{displaymath} (7)

v={du\over dr}={A\over r}
\end{displaymath} (8)

du = A{dr\over r},
\end{displaymath} (9)

so the solution is
u=A\ln r.
\end{displaymath} (10)

Ignoring the trivial additive and multiplicative constants, the general pure radial solution then becomes
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} (11)

Other solutions may be obtained by differentiation, such as

$\displaystyle u$ $\textstyle =$ $\displaystyle {x-a \over (x-a)^2+(y-b)^2}$ (12)
$\displaystyle v$ $\textstyle =$ $\displaystyle {y-b \over (x-a)^2+(y-b)^2},$ (13)

$\displaystyle u$ $\textstyle =$ $\displaystyle e^x\sin y$ (14)
$\displaystyle v$ $\textstyle =$ $\displaystyle e^x\cos y,$ (15)

\tan^{-1}\left({y-b\over x-a}\right).
\end{displaymath} (16)

Harmonic functions containing azimuthal dependence include
$\displaystyle u$ $\textstyle =$ $\displaystyle r^n\cos (n\theta)$ (17)
$\displaystyle v$ $\textstyle =$ $\displaystyle r^n\sin (n\theta).$ (18)

The Poisson Kernel
u(r,R,\theta,\phi)={R^2-r^2\over R^2-2rR\cos(\theta-\phi)+r^2}
\end{displaymath} (19)

is another harmonic function.

See also Scalar Potential, Vector Potential


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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© 1996-9 Eric W. Weisstein