# EM basis functions

Mentor
2021 Award
Are there any sets of basis functions that are particularly useful for Maxwell's equations? I was thinking about Fourier just because it is the first basis I always think of, but I don't know that it would actually be a convenient basis. For example, I don't know that curl or divergence would be easy in the Fourier basis.

marcusl
Gold Member
"Maxwell's Equations" are pretty open ended. There are bases that are useful for solving certain problems.
1. For scalar potential and Green function problems in magneto- and electro-statics, cylindrical and spherical harmonics come to mind as particularly useful. There are two different cylindrical harmonic bases as well, those with cosine/sines and ordinary Bessels, and those with cosh/sinh and modified Bessels. There are different versions of the spherical expansions, too. The alternate expansions sometimes have particular advantages, see
http://physics.princeton.edu/~mcdonald/examples/EM/cohl_apj_527_86_99.pdf" [Broken]
and
http://iopscience.iop.org/0022-3727/19/8/001?ejredirect=migration"
for two interesting applications.
2. Similar harmonics exist of course for spheroidal and all the other 8 or so coordinate systems that are separable under Laplace's equation.
3. The harmonic bases are useful for radiation problems as well (e.g., multipoles). For full EM wave problems that include polarization, the vector forms of the harmonic bases are used. Vector spherical harmonics are the best known of these.
4. Sines and cosines (your Fourier bases) are useful for problems in rectangular coordinates.

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Spin functions seem to be basis functions of the equations. I've heard it expressed that quaterions can be reduced to a pair of complex-valued 2 component spinors. And quaterions are the most reduced form of transformations for SR. Any combination of rotations or boosts, inclusive or exclusive, can be performed most simply with spinors and quaterions.

Since SR is a particular analysis of Maxwell's Equations regarding time and length (minus the conjectural things Einstein added) spinors and quaterions must be the most reduced form of transformations of variables with the Maxwell Equations.

Quaterions function in that way apparently because they encode symmetry properties which vectors do not. At least that is my understanding.

I happen to use the vector spherical harmonics recently. They are the basis functions to describe the electric and magnetic field in the spherical coordinate. However, I find the derivation of the vector spherical harmonics in Jackson's a little awkward. Bohren's book on absorption and scattering of light by small particles has a different derivation I find easier to chew.

Mentor
2021 Award
Thanks for the replies so far! I looked into Fourier a bit more and I found that divergence and curl can be represented by reasonably straightforward operations on the Fourier domain, but then I realized that what is really needed is a basis where the inverse of the curl and divergence has a simple representation. Unfortunately, I got stuck there. I will look into the details of what you have all posted over the next day or so.