Hacker News Reader

Hacker News

Comment by dfdz | Hacker News Reader

Parent Full thread

dfdz

The key point which the author eventually realizes:

> Circular Harmonics are just a Fourier series

The term circular harmonics is just a nonstandard name for Fourier series.

View on HN

We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started

And know the place for the first time.

T. S. Eliot - Little Gidding

I think the standard name is "Besell Functions" https://en.wikipedia.org/wiki/Bessel_function

Not really. In Mathematics "circle" refers to a boundary of a disk. So it's a 1D periodic manifold, and the circular harmonics/Fourier series are the basis for such periodic functions. The bessel functions give a basis for the radial part of 2D solutions to a certain class of differential equations on a 2D disk. Incidentally the radial part of those solutions are circular harmonics.

And there are many. Zernike polynomials are a common one.

Not a wizard or anything but reading wikipedia:

Bessel functions for integer α are also known as cylinder functions or the cylindrical harmonics

Bessel functions are useful when you must solve the volume of the cylinder (or the volume outside a cylinder). If you only care about the surface of the cylinder, you must use the Fourier series in one direction and the Fourier transform in the other direction.

I (GGP comment) was wrong.

You are right.

No. Bessel functions are the solution to many differential equations posed in cylindrical coordinates.

Legendre polynomials are what we are seeing here which are solutions to many differential equations posed in spherical coordinates.

Classical sine/cosine Fourier solutions are the solutions to many differential equations posed in cartesian coordinates.

So these spherical solutions should also remind people of atomic quantum "orbital" shapes because those are solutions to Schrödinger's (differential) equations in spherical coordinates.

Actually it's a Fourier series (or even a Laplace transform) with a coordinate frame change - spherical instead of cartesian (another is cylindrical coordinates).

Normal everyday stuff for EEs and Physicists.

Reply on news.ycombinator.com