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- Thread starter kthouz
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Meir Achuz

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It sounds just like cos and sin to me.

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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.

There they look like modified cosines.

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HallsofIvy

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[itex]GL_n(r) = L_n(r^2) exp(-r^2/2)[/itex]

where [itex]L_n(x)[/itex] 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.

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

or, if you prefer real basis,

[tex]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}.[/tex]

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

[tex]\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).[/tex]

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

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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].

See the end of http://faculty.uml.edu/cbaird/95.657%282011%29/EMLecture4.pdf" [Broken].

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Perhaps Zernike polynomials?

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