2026-04-28 23:37 UTC
@johndcook@mathstodon.xyz the approximation you had derived in that post is also what you get when you try expanding the Bessel function into a series of cosines, using Bürmann's theorem (https://mathworld.wolfram.com/BuermannsTheorem.html) and then take the first two terms. \[ J_0(x)=1+\frac{\cos(x)-1}{2}-\frac{(\cos(x)-1)^2}{48}+\frac{7 (\cos(x)-1)^3}{1440}+\cdots\]
Replies (1)
-
@johndcook@mathstodon.xyz 2026-04-30 21:56
@tpfto@mathstodon.xyz That's really cool. Hadn't seen that before. Maybe I'm missing something. In the notation of the Mathworld article, a = 0 and phi = cos. But phi'(a) = 0, and so it seems Bürmann's theorem doesn't apply.