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# Reproducing T511
The table holds 88 values of the standard DLMF irregular Coulomb function G.
Its title, parameter order, sign and normalization agree with family #210.
The parameters are ell (nonnegative integer), eta (rational), rho (positive
rational). Its initial finite selection is ell=0..3; eta=0,+/-1/2,+/-1,+/-2;
rho=1/2,1,2; and additionally rho=5 at eta=2. Entries are ordered with the
free case first, then increasing absolute coupling, positive before negative.
Install numberdb in Sage, then download generate.py and checks.py beside this
file. Run `sage -python checks.py` for every local numerical check. Run
`sage -python generate.py` for the client comparison with every stored value.
Run `sage -python checks.py --stored` to repeat the independent checks on the
actual database decimals and their exact uncertainty intervals. Reading the
unpublished draft requires NUMBERDB_API_KEY. The local check needs no key.
The optional `sage -python generate.py --fill-draft` sends entries and the
four declared attachments. It needs all four files and a key authorized to
write T511. It does not publish or offer the draft for review. This build
uses a complete document submission after the private checks, so the numbers
and final mathematical prose arrive together; the package checks that the
stored numbers agree with this generator.
The initial precision, 515 bits, was measured, not inferred from a decimal
working-precision setting. The least accurate value retained 496 bits, and
all balls fit strictly within the 100-significant-digit displayed intervals.
The mathematical rigour is supplied by Arb, not by wrapping mpmath values.
The independent mpmath comparisons at 150 and 200 digits are additional checks.
Exact versions and numerical-check counts are in evidence.json.
Handbook control: Abramowitz and Stegun, page 543, Example 2, includes
G_0(2,5)=0.79445 and G'_0(2,5)=-0.67049, and gives G_1..G_3 as
1.0815, 1.4969, 2.0487, with -G'_1..-G'_3 as 0.60286, 0.56619, 0.79597.
The printed intermediate rounding warrants the explicit absolute tolerance
0.0001, rather than a claim that every printed last digit is certified.
The separate mpmath documentation prints more accurate controls for ell=1..3.
The handbook scan was inspected directly.
Relevant sources:
- https://dlmf.nist.gov/33.2 (definition, sign, normalization)
- https://dlmf.nist.gov/33.4 (recurrences and derivatives)
- https://dlmf.nist.gov/33.5 (free Bessel specialization)
- https://dlmf.nist.gov/33.10 (asymptotic convention)
- https://mpmath.readthedocs.io/en/stable/functions/bessel.html#coulomb-wave-functions
- https://personal.math.ubc.ca/~cbm/aands/page_543.htm
The proposed quantum mechanics tag is absent from the live list. The table
uses existing special functions, special values and physics tags. A quantum
mechanics tag would also connect T510, T404, T406, T408 and T411. Tag creation
is left for the separate editorial decision.