j0 (and y0) in the range 2 <= x < (p/2)*log(2)

Steve Kargl sgk at troutmask.apl.washington.edu
Tue Sep 4 04:10:13 UTC 2018

On Tue, Sep 04, 2018 at 03:56:28AM +0000, Montgomery-Smith, Stephen wrote:
> A quick google search turned up this
> https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf
> which has the functions p0 and q0.  Maybe this was the basis of this code.

I've read that paper.  It uses |x| > 45 for the cut over 
to the large argument asymptotic expansion.  One of the
primary results for that paper is the development of 
new approximations that are robust near zeros of Jn(x).
In the the discussion of the results, the paper notes
the use of a double-double representation for intermediate

A&S claims that the remainder in truncating the series
does not exceed the magnitude of the first neglected
term.  If you set x = 2 and compute the terms in 
p0(x), one finds the smallest term is about |pk| = 1e-4.


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