Oh, this is something I use all the time! It turns out if you use complex numbers for your coefficients, then you can represent any analytic 2D function with c

Oh, this is something I use all the time! It turns out if you use complex numbers for your coefficients, then you can represent any analytic 2D function with circular harmonics, like this: F(x,y) = ∑ Cₙ (x + iy)ⁿ⁻¹ It turns out many real, useful fields satisfy this condition, such as heat flow, electric and magnetic fields, fluid flow and more. Circular harmonics like this are used extensively in the design, testing, and running of particle accelerators. The magnets that control the particle beams are often modelled as having magnetic fields described by 2D circular harmonics. The equation above can be used to plot something like this: https://i.imgur.com/o9nUmVr.png https://i.imgur.com/o9nUmVr.png That plot was generated from 21 complex numbers, for example. In fact, people build machines that directly measure these coefficients by rotating wires in a magnetic field: https://jsss.copernicus.org/articles/9/99/2020/ https://jsss.copernicus.org/articles/9/99/2020/