An edited online reference published by National Institute of Standards and Technology. The authorship tags on individual chapters refer to the credited chapter contributors, not to authorship of the entire reference.
Official text: dlmf.nist.gov/3.8
Relevant locators: §3.8(i), equation 3.8.3 and following definitions of local order of convergence; §3.8(iii), equation 3.8.6 and the Secant Method paragraph, which distinguishes the secant method from bracket-preserving alternatives. The derivations and calculations in the solutions are given independently.
Official NIST DLMF section 5.12. Equation 5.12.1 gives Euler's integral for the beta function and its expression as a ratio of gamma functions, with positive real parts of the two parameters. The integral identity supplies normalization and shifted-parameter moments for the beta distribution.
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.
Online text: dlmf.nist.gov/3.5#i
Cited passages: NIST Digital Library of Mathematical Functions, §3.5(i), equation 3.5.1 and its C² hypothesis; the source writes integral minus rule; NIST Digital Library of Mathematical Functions, §3.5(i), equations 3.5.2–3.5.3: composite formula and signed error.
Cited passages: NIST Digital Library of Mathematical Functions, §3.5(iv), equation 3.5.15 and the following discussion of nodes and weights.
Online text: dlmf.nist.gov/24
Online text: dlmf.nist.gov/25.6
Part of Chapter 25, Zeta and Related Functions, credited to T. M. Apostol. The site release date is not treated as the chapter publication year.
Cited passages: §25.6(i), equations 25.6.1–25.6.2: positive even arguments.
Online text: dlmf.nist.gov/27.9
Part of Chapter 27, Functions of Number Theory, credited to T. M. Apostol. The site version release date is not treated as the chapter publication year.
Online text: dlmf.nist.gov/2.5
Online text: dlmf.nist.gov/5.2
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.
Online text: dlmf.nist.gov/7.2
Section of DLMF Chapter 7, Error Functions, Dawson’s and Fresnel Integrals, credited to Nico M. Temme on dlmf.nist.gov/7. The inspected site release is version 1.2.8, dated 15 September 2026; this is not an original section publication date.
Cited passages: NIST Digital Library of Mathematical Functions, §7.2, equations 7.2.1–7.2.4; §7.2(i), equation 7.2.1, entire-function statement and real-axis limits in 7.2.4; §7.2(i), equation 7.2.2 and the entire-function statement.
Online text: dlmf.nist.gov/7.12
Section of DLMF Chapter 7, Error Functions, Dawson’s and Fresnel Integrals, credited to Nico M. Temme on dlmf.nist.gov/7. The inspected site release is version 1.2.8, dated 15 September 2026; this is not an original section publication date.
Cited passages: §7.12(i), equation 7.12.1 and its real-axis remainder statement.
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]
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.
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]
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.
Section of NIST Digital Library of Mathematical Functions, Chapter 3, Numerical Methods, whose credited chapter author is Nico M. Temme (N. M. Temme on the chapter landing page). This credit concerns Chapter 3, not the whole edited collection. The National Institute of Standards and Technology issues and hosts the reference. No separate section publication or revision date is established.
Online text: dlmf.nist.gov/3.3
Chapter authorship: dlmf.nist.gov/3
Cited passages: NIST Digital Library of Mathematical Functions, Chapter 3, §3.3 Interpolation: Chapter 3 landing page: N. M. Temme byline and acknowledgements distinguish author from contributors; §3.3(iii), equation 3.3.34: distinct-node base cases and recurrence; §3.3(iii), equation 3.3.35: explicit symmetric sum; §3.3(iv), equation 3.3.38 and following new-node update paragraph.
Section of NIST Digital Library of Mathematical Functions. Chapter 3 author: Nico M. Temme, credited as N. M. Temme. National Institute of Standards and Technology issues and hosts the edited reference. The inspected online release is version 1.2.8, 15 September 2026; this is a site release date, not a section publication date. No separate subsection publication or revision date is established.
Chapter credit: dlmf.nist.gov/3
Cited passages: NIST Digital Library of Mathematical Functions, §3.5(ii), equations 3.5.6–3.5.8.
Chapter authors: Tom H. Koornwinder, Roderick S. C. Wong, Roelof Koekoek, René F. Swarttouw, William P. Reinhardt. These are Chapter 18 credits in the edited NIST Digital Library of Mathematical Functions. The National Institute of Standards and Technology issues and hosts the reference. No chapter publication date is established; version 1.2.8, released 15 September 2026, is an online reference release.
Chapter and author credits: dlmf.nist.gov/18

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