# Circular Harmonics

I have a problem consisting in solving for potential in 2 dim polar coordinates where I am asked to use circular harmonics. Can I still use Legendre polynomials (since these are actually for spherical harmonics)? If not what are their analoguous in polar coordinates?

Meir Achuz
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It sounds just like cos and sin to me.

Meir Achuz
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Circular harmonics are discussed at <http://www.blackpawn.com/texts/ch/default.html>. [Broken]
There they look like modified cosines.

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HallsofIvy
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The Laplacian, in polar coordinates, reduces to Bessel's equation. Bessel functions are harmonics for two dimensional circles (or three dimensional cylinders).

I believe that "circular harmonics" refers to the Gauss-Laguerre functions defined by

$GL_n(r) = L_n(r^2) exp(-r^2/2)$

where $L_n(x)$ is the nth Laguerre polynomial. (They are often denoted by a script L, but I don't know how to do that.) They turn up in descriptions of circular radar and laser beams. They constitute a complete orthogonal basis and have the very handy property that each is its own Hankel transform. They're distinctly different from both Legendre polynomials and Bessel functions, but they're useful for expanding potentials in circular geometries.

vanhees71
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The Bessel and Neumann (or equivalently the Hankel) functions provide a basis for the radial equation, when separating the 2D Laplace equation in polar coordinates. For the angular part, which I'd identify with the analog of spherical harmonics in 2D are simply the orthonormal set of exponential functions

$$u_m(\varphi)=\frac{1}{\sqrt{2 \pi}} \exp(\mathrm{i} m \varphi), \quad m \in \mathbb{Z},$$

or, if you prefer real basis,

$$u_0(\varphi)=\frac{1}{\sqrt{2 \pi}}, \quad u_m^{(1)}(\varphi)=\cos(m \varphi), \quad u_m^{(2)}(\varphi) \sin(\varphi), \quad m \in \mathbb{N}_{>0}.$$

The general solution of the Laplace equation in terms of the corresponding series reads

$$\phi(r,\varphi)=\sum_{m=-\infty}^{\infty} [\phi_m^{(1)} J_m(r/r_0) + \phi_m^{(2)} N_m(r/r_0)] u_m(\varphi).$$

This form with the Bessel and Neumann functions is convenient since $J_m$ is the solution of the radial equation which is analytic in $r=0$, while $N_m$ is singular at the origin.

The OP specifically said two-dimensional. The solution to the potential in two-dimensional polar coordinates is not Bessel functions (that is the solution to three-dimensional cylindrical coordinates) but us just powers of the radial coordinate times sines and cosines of the angular coordinate. In two-dimensions, circular harmonics are just sines and cosines in the angular coordinate with a single-valued condition applied (thus leading to harmonics).
See the end of http://faculty.uml.edu/cbaird/95.657%282011%29/EMLecture4.pdf" [Broken].

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Andy Resnick