Harmonic Function

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

$$\nabla^2 u(x,y)=0$$ (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

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

and consider only radial solutions

$$u_{rr}+{1\over r} u_r=0.$$ (3)

This is integrable by quadrature, so define $v\equiv du/dr$,

$${dv\over dr}+{1\over r}v=0$$ (4)

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

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

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

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

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

so the solution is

$$u=A\ln r.$$ (10)

Ignoring the trivial additive and multiplicative constants, the general pure radial solution then becomes

$$u=\ln[(x-a)^2+(y-b)^2]^{1/2} = {\textstyle{1\over 2}}\ln\left[{(x-a)^2+(y-b)^2}\right].$$ (11)

Other solutions may be obtained by differentiation, such as

$$u = {x-a \over (x-a)^2+(y-b)^2}$$ (12)

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

$$u = e^x\sin y$$ (14)

$$v = e^x\cos y,$$ (15)

and

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

Harmonic functions containing azimuthal dependence include

$$u = r^n\cos (n\theta)$$ (17)

$$v = r^n\sin (n\theta).$$ (18)

The Poisson Kernel

$$u(r,R,\theta,\phi)={R^2-r^2\over R^2-2rR\cos(\theta-\phi)+r^2}$$ (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.