For my own future reference, in case I lose the bit of paper on which I jotted it.
In a function of period 2π, a unit step discontinuity in the nth derivative at phase α contributes this to the Fourier series:
i (in e-ik(t-α) – i-n eik(t-α)) / (2πkn+1)
I haven’t the skill to prove this for general n, but then I’m unlikely to need it for n>2.
In other words:
Each of these is the derivative of the next, so it appears to be correct by induction. Yet somehow I am not fully confident of my reasoning.
Did I ever work out the series for n=2? Cases 1 and 0 came up in some of my experiments in curve-fitting (not blogged), but I do not now recall if I had any use for n=2.
Heh, a chain of tangent thoughts led from that question to a new approach to my stalled font-fitting project.