Apropos Watson's Treatise (I didn't manage to cover this with Neil Wednesday, so now everybody gets it, even though most of you understand it better than I do): Early on, (p4 of 804) Watson recounts Euler's analysis of the motion of a hanging chain, which is further modernized (w/o attribution) at http://mysite.du.edu/~jcalvert/math/hchain.htm . I was a bit disappointed that this classic solution makes *two* small-deflection approximations: The chain particles move only horizontally, and only sinusoidally. In this solution, the amplitude varies along 0 < x < length in proportion to J[0](2 ω sqrt (x/g)) at one of a discrete set of angular velocities (frequencies/2π) ω determined by J[0](2 ω sqrt (length/g)) = 0 (reflecting the fixed top end). Thus, to learn the possible frequencies and nodes, we need the roots of J[0](sqrt X). (No less than) Watson: "By an extremely ingenious analysis, ..., Euler proceeded to shew that the three smallest roots of [J[0](2 sqrt x)] are 1.4457965, 7.6178156, and 18.7217517." Should be 1.445796490736696, 7.617815585915522, 18.7217516976738, but this I had to see. Euler's idea is similar to his zeta(2) conjecture, and works for functions where you have the power series, and the sum of the reciprocated roots σ[1] := Σ_k r[k]^-1 converges. (E.g., sinc(sqrt x).) Briefly: Assume you can write f(x) as a Weierstrass product f(0) prod(1-x/r[n]), with 0 < r[n] < r[n+1]. Expanding the log derivative of the product and power series gives a system of linear equations that can be solved for σ[n] := Σ_k r[k]^-n, n=1,2,... . Then the inequality σ[n]^-n < r[1] < σ[n]/σ[n+1] gets r[1]≈1.445795 when n reaches 7. Writing σ'[n]:=σ[n]-r[1]^-n gives the inequality σ'[n]^-n < r[2] < σ'[n]/σ'[n+1], &c. --rwg PS, a few months ago, under the Subject: Timber!, I sent a faulty formula for a toppling pole, followed by a correction. Here's a graph<http://gosper.org/topple.png>of initial tilt vs toppling time. On Mon, Apr 2, 2012 at 5:11 PM, Bill Gosper <billgosper@gmail.com> wrote:

*Quad>*

Hey all.

Does anyone have any ideas on computing Bessel functions to high precision (over the complex plane)?

Googling gives [1] and [2]. Sorry, [1] is behind a paywall.

I am wondering if one of you has a secret magical sum or product that just magically works well for high precision computation. If not, maybe we can develop some strategies!

Thanks,

Robert Smith

[1] dl.acm.org/citation.cfm?id=1466254 [2] web.cs.dal.ca/~jborwein/bessel.pdf <http://web.cs.dal.ca/%7Ejborwein/bessel.pdf>

*------ [...] At the time, I used (a tiny fraction of) Watson's Treatise, but now there's dlmf*.nist.gov/10 .[...] --rwg