Post #2649075 · Elektrine

2026-04-30 21:56 UTC

@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.

Replies (2)

@johndcook@mathstodon.xyz 2026-04-30 22:11

@tpfto@mathstodon.xyz Looks like you can generalize Burmann's theorem by generalizing the series inversion theorem. This article includes the case of phi having a fixed number of zero derivatives at a. https://www.mathematica-journal.com/2014/11/24/on-burmanns-theorem-and-its-application-to-problems-of-linear-and-nonlinear-heat-transfer-and-diffusion/

Open ##2649076
@tpfto@mathstodon.xyz 2026-04-30 22:29

@johndcook@mathstodon.xyz Yes, in that particular case, you actually get an indeterminate form, so you have to take limits after doing the expansion for generic a. This was fresh on my mind when I read your blog entry, since I had recently figured out a slightly more tractable implementation of Bürmann (at least, compared to repeatedly differentiating that divided difference in the original formula).

Open ##2649077