Date: Mon, 7 Oct 96 23:57:16 CDT
From: rusin (Dave Rusin)
To: don, thunder, wu
Subject: I didn't make them up
While standing around yakking Monday I mentioned some results I was
sure I had once known but forgotten. Senility being what it is, I already
can't remember quite what I said. But I did do some searching just now
and am reminded of what little I did once know:
The Arithmetic-Geometric mean can indeed be used to calculate certain
elliptic integrals. It's an iterative process, replacing (a,b) with
((a+b)/2, sqrt(ab)). I don;t remember quite what it converges to
except that with suitable (a,b) one gets extremely rapid convergence
to, say, pi. If my notes are correct, there was a paper by the
Borweins on this in the Intelligencer about 5 years ago, and in
Scientific American a little before that. ("Rapid" means, e.g., the
number of correct digits more than doubles each time.)
The "edge-of-the-wedge" theorem is actually a family of results typically
asserting that a map of germs of functions (comparatively smooth) is onto.
The original theorem, as far as I can tell from a quick web search, was
due to Bogolyubov: a function holomorphic on a wedge is holomorphic in
a nbhd of the origin. (A wedge is a product of upper half planes.) I think
I'm missing some key ingredients here, but for my purposes it's enough
to remember "it's a little like Hartog's theorem".
I had mentioned this in the context of theorems with a great name. I
thought of a few others: the Snake Lemma, the Pigeonhole Principle, and
You Can('t) Hear The Shape Of A Drum. Sometimes names are chosen which
convey the idea immediately, even when you've never heard the name before.
Sort of like Stirrup Pants and the Car Bra.
dave