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1: 25.10 Zeros
The product representation (25.2.11) implies ζ ( s ) 0 for s > 1 . …The functional equation (25.4.1) implies ζ ( 2 n ) = 0 for n = 1 , 2 , 3 , . … is chosen to make Z ( t ) real, and ph Γ ( 1 4 + 1 2 i t ) assumes its principal value. …Because Z ( t ) changes sign infinitely often, ζ ( 1 2 + i t ) has infinitely many zeros with t real. … More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …
2: Bibliography B
  • G. E. Barr (1968) A note on integrals involving parabolic cylinder functions. SIAM J. Appl. Math. 16 (1), pp. 71–74.
  • R. L. Bishop (1981) Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).
  • G. Blanch and D. S. Clemm (1962) Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..
  • H. M. Bui, B. Conrey, and M. P. Young (2011) More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (1), pp. 35–64.
  • P. G. Burke (1970) A program to calculate a general recoupling coefficient. Comput. Phys. Comm. 1 (4), pp. 241–250.
  • 3: 16.17 Definition
    §16.17 Definition
    Then the Meijer G -function is defined via the Mellin–Barnes integral representation: … When more than one of Cases (i), (ii), and (iii) is applicable the same value is obtained for the Meijer G -function. Assume p q , no two of the bottom parameters b j , j = 1 , , m , differ by an integer, and a j b k is not a positive integer when j = 1 , 2 , , n and k = 1 , 2 , , m . Then …
    4: Possible Errors in DLMF
    One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the [Uncaptioned image] icon) for links to defining formula. …Errors in the printed Handbook may already have been corrected in the online version; please consult Errata. …
    5: Errata
  • Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions

    Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, 𝖯 n , 𝖰 n , P n , Q n , 𝑸 n and the Laguerre polynomial, L n , were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

  • Subsection 25.10(ii)

    In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).

  • Equation (3.3.34)

    In the online version, the leading divided difference operators were previously omitted from these formulas, due to programming error.

    Reported by Nico Temme on 2021-06-01

  • Table 22.4.3

    Originally a minus sign was missing in the entries for cd u and dc u in the second column (headed z + K + i K ). The correct entries are k 1 ns z and k sn z . Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions sn , cn , dn , whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

    u
    z + K z + K + i K z + i K z + 2 K z + 2 K + 2 i K z + 2 i K
    cd u sn z k 1 ns z k 1 dc z cd z cd z cd z
    dc u ns z k sn z k cd z dc z dc z dc z

    Reported 2014-02-28 by Svante Janson.

  • References

    Bibliographic citations were added in §§3.5(iv), 4.44, 8.22(ii), 22.4(i), and minor clarifications were made in §§19.12, 20.7(vii), 22.9(i). In addition, several minor improvements were made affecting only ancilliary documents and links in the online version.

  • 6: Foreword
    In 1964 the National Institute of Standards and Technology11 1 Then known as the National Bureau of Standards. published the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. … The online version, the NIST Digital Library of Mathematical Functions (DLMF), presents the same technical information along with extensions and innovative interactive features consistent with the new medium. …
    7: 3.12 Mathematical Constants
    3.12.2 π = 4 0 1 d t 1 + t 2 .
    For access to online high-precision numerical values of mathematical constants see Sloane (2003). …
    8: How to Cite
    The direct correspondence between the reference numbers in the printed Handbook and the permalinks used online in the DLMF enables readers of either version to cite specific items and their readers to easily look them up again — in either version! The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online. …
    9: About the Project
    Refer to caption
    Figure 1: The Editors and 9 of the 10 Associate Editors of the DLMF Project (photo taken at 3rd Editors Meeting, April, 2001). …
    These products resulted from the leadership of the Editors and Associate Editors pictured in Figure 1; the contributions of 29 authors, 10 validators, and 5 principal developers; and assistance from a large group of contributing developers, consultants, assistants and interns. … They were selected as recognized leaders in the research communities interested in the mathematics and applications of special functions and orthogonal polynomials; in the presentation of mathematics reference information online and in handbooks; and in the presentation of mathematics on the web. …
    10: Bibliography U
  • H. Umemura and H. Watanabe (1998) Solutions of the third Painlevé equation. I. Nagoya Math. J. 151, pp. 1–24.
  • Unpublished Mathematical Tables (1944) Mathematics of Computation Unpublished Mathematical Tables Collection.
  • J. Urbanowicz (1988) On the equation f ( 1 ) 1 k + f ( 2 ) 2 k + + f ( x ) x k + R ( x ) = B y 2 . Acta Arith. 51 (4), pp. 349–368.