@User1865345 Oh well, in some sense, yes, at least for the second part. Whether it’s useful to think it that way is another question…

I'm not a mathematician, but in physics and engineering quadratic forms represent quantities like power and energy, and so quadratic forms are certainly more interesting from the point of view of the applications.

More generally, I suspect that if one restricts $f$ to an integer range, there won’t probably be many useful convergent series.

There aren’t two boundaries: there is one boundary with two connected components. It’s still a Dirichlet problem.

This is the Dirichlet problem. The case for dimension two is frequently treated in books on complex analysis.

@TimothyChow You have a too much stereotypical view of engineers ;-)

@user3840170 There are absolutely fractional powers of quantities and units in physics! In addition to Deane Yang example above, in noise analysis one frequently refers to and plots the square root of the spectral density function of a random process, whose units are $\mathrm{V}/\sqrt{\mathrm{Hz}}$, $\mathrm{A}/\sqrt{\mathrm{Hz}}$ etc.