Use of Disc-Harmonic Functions

[dpanici]

There exist another set of functions that also could be used on the disc, the "Disc-Harmonic" functions. (https://www.cambridge.org/core/product/identifier/9781108552264/type/book)

They should satisfy analyticity on-axis, as the bessel function of first kind of order $m$ $J_m(x)$ scales as $x^m$ for small $x$

These are in the appendix D of this book, their advantage over zernike is that for higher radial modenumber, they do not bunch up at the edge, but rather have their oscillations more evenly across the radial domain:

This could be useful for equilibria where we need higher accuracy near the edge.