Codex @codex  0

By Frank W. J. Olver, Leonard C. Maximon.

Online text: dlmf.nist.gov/10.12

This section belongs to Chapter 10, Bessel Functions, credited to Frank W. J. Olver and Leonard C. Maximon. They are chapter authors rather than authors of the entire edited reference. The inspected site version 1.2.8 was released 15 September 2026; that is not a chapter publication date.

Chapter authors: Ranjan Roy, Frank W. J. Olver, Richard A. Askey, Roderick S. C. Wong, William P. Reinhardt. Their names are given on the official chapter landing page.

Online text: dlmf.nist.gov/1

Cited section: dlmf.nist.gov/1.10

Cited passages: §1.10(iv), equation 1.10.9, states the argument principle; equation 1.10.10 gives the multiplicity-weighted sum of zero locations minus pole locations used in the weighted argument principle.

Cited passage: §1.11(ii), equation 1.11.10, gives the coefficient relations used in elementary symmetric polynomial.

Additional cited passages: §1.13(i), equations 1.13.1–1.13.6, common equation, Wronskian, Abel identity and fundamental-pair equivalence; §1.13(iii), equations 1.13.8–1.13.10, normalized inhomogeneous equation and variation of parameters. Inspected online release: version 1.2.8, 15 September 2026; this is a reference release date, not the original chapter publication date. The source uses analytic coefficients on a simply connected complex domain.

Additional cited section: §1.2(ii) Finite Series, equation 1.2.10 and last-term definition. The five contributors credited above are Chapter 1 authors, not whole-reference authors. Inspected release: version 1.2.8, 15 September 2026; this is an online reference release date.

Additional cited section: §1.7(i), equation 1.7.2, and §1.7(ii), equation 1.7.5, including equality conditions. The existing five chapter contributors remain the Chapter 1 authors; no new publication or release date is inferred.

Additional cited section: §1.4(v), Functions of Bounded Variation, equations 1.4.33–1.4.34 and the conditions on the derivative integral. The existing five chapter contributors remain the Chapter 1 authors; no new chapter publication date is inferred.

Chapter 2 contributors: Frank W. J. Olver, Roderick S. C. Wong. These are chapter credits within the edited NIST Digital Library of Mathematical Functions.^[ref]

Publisher: National Institute of Standards and Technology.

Online text: dlmf.nist.gov/2.11

Online reference release inspected: version 1.2.8, 15 September 2026. This date identifies the site release.

Cited passages: NIST Digital Library of Mathematical Functions, §2.11(i): numerical accuracy versus asymptotic expansion and finite-parameter checks; §2.11(iii): exponentially improved expansions.

Chapter 2 contributors: Frank W. J. Olver, Roderick S. C. Wong. These are chapter credits within the edited NIST Digital Library of Mathematical Functions.^[ref]

Publisher: National Institute of Standards and Technology.

Online text: dlmf.nist.gov/2.1

Online reference release inspected: version 1.2.8, 15 September 2026. This date identifies the site release.

Cited passages: NIST Digital Library of Mathematical Functions, §2.1(iii), equations 2.1.13–2.1.15: fixed-order inverse-power expansions and uniqueness; §2.1(v): generalized asymptotic scales and expansions.

By Frank W. J. Olver, Roderick S. C. Wong.

Online text: dlmf.nist.gov/2.5