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# Positive zeros of the Kelvin function ber_0 (T589)
Definition fixed before computation: x_n is the nth strictly positive real
zero of ber_0(x) = Re J_0(x exp(3 pi i/4)) = Re I_0(x exp(pi i/4)),
in increasing order, starting at n=1. The value is the dimensionless x itself,
not x/sqrt(2), x squared, a function value, or a derivative zero. J_0 and I_0
are entire; there is no branch convention left at integer order zero.
The only free parameter is n, an integer with n >= 1. Its selection category
is discrete_index: the roots are intrinsic features of a named function.
Enumerate every n=1,...,20 in order. The cutoff is a noncanonical finite
reference depth, giving several oscillations beyond the first few roots while
allowing exhaustive root counting and independent 100-digit checks. It is not
a claim that each of the twenty radii has its own published application.
The fixed order zero is the standard ber function documented in DLMF 10.61
and 10.65, not a sample of a continuous order parameter. DLMF 10.73(iii)
documents these Kelvin functions in circular-conductor current distributions.
This is context for the family; this table does not assert that individual
ber zeros are the eigenvalues of every plate or diffusion boundary problem.
DLMF 10.75(xii) identifies a twenty-root table in Zhang and Jin (1996), p. 323,
which is a reference-coverage precedent. That book page was not consulted.
The first five ber roots were read directly from the scanned A&S p. 381.
Breadth was considered before depth. The proposal family treats the regular
pair ber/bei and the decaying pair ker/kei in separate tables. Within this
named object the consecutive first twenty positive roots have no gaps; adding
higher angular orders or derivative roots would change the scope. Increasing
one component to hundreds of roots would sacrifice exhaustive checking for
little additional reference coverage. There are no continuous reference
arguments, rational meshes, mixed roles, hidden couplings, or Cartesian products.
The rational scan points used by the certificate are computational partition
endpoints, not selected table arguments.
The parameter constraint is n >= 1. The completeness note states that all
first twenty positive order-zero roots are present, why this depth was chosen,
and that the infinite family is incomplete. The default generator enumeration,
Programs example, and stored rows use this same selection.
Sources checked in this run:
- https://dlmf.nist.gov/10.61 (10.61.1 and 10.61.3): definition and equation.
- https://dlmf.nist.gov/10.65 (10.65.2): real power series and origin value.
- https://dlmf.nist.gov/10.68 (10.68.1 and 10.68.3): modulus and phase.
- https://dlmf.nist.gov/10.71 (10.71.8): Wronskian integral identity.
- https://dlmf.nist.gov/10.73#iii: physical context, not per-root demand.
- https://dlmf.nist.gov/10.75#xii: finite root-reference precedent.
- https://personal.math.ubc.ca/~cbm/aands/page_381.htm: five printed controls.
Policy evidence: ../20261028T195045Z-build-skills/manifest.json, archived
local and live skill SHA-256 (both):
e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.
This builder's justification is not the independent critique acceptance.