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1: 27.13 Functions
§27.13(iv) Representation by Squares
2: 22.16 Related Functions
See Figure 22.16.2. …
3: Bibliography G
  • E. Grosswald (1985) Representations of Integers as Sums of Squares. Springer-Verlag, New York.
  • 4: Bibliography E
  • T. Estermann (1959) On the representations of a number as a sum of three squares. Proc. London Math. Soc. (3) 9, pp. 575–594.
  • 5: 1.17 Integral and Series Representations of the Dirac Delta
    §1.17 Integral and Series Representations of the Dirac Delta
    §1.17(ii) Integral Representations
    Then comparison of (1.17.2) and (1.17.9) yields the formal integral representation
    Coulomb Functions (§33.14(iv))
    §1.17(iii) Series Representations
    6: Bibliography M
  • L. J. Mordell (1917) On the representation of numbers as a sum of 2 r squares. Quarterly Journal of Math. 48, pp. 93–104.
  • 7: Bibliography
  • A. Apelblat (1991) Integral representation of Kelvin functions and their derivatives with respect to the order. Z. Angew. Math. Phys. 42 (5), pp. 708–714.
  • T. M. Apostol and H. S. Zuckerman (1951) On magic squares constructed by the uniform step method. Proc. Amer. Math. Soc. 2 (4), pp. 557–565.
  • R. Askey and J. Fitch (1969) Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appl. 26 (2), pp. 411–437.
  • R. Askey (1974) Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal. 5, pp. 119–124.
  • 8: 26.15 Permutations: Matrix Notation
    The sign of the permutation σ is the sign of the determinant of its matrix representation. The inversion number of σ is a sum of products of pairs of entries in the matrix representation of σ : … Let r j ( B ) be the number of ways of placing j nonattacking rooks on the squares of B . …
    9: Errata
  • Chapter 19

    Factors inside square roots on the right-hand sides of formulas (19.18.6), (19.20.10), (19.20.19), (19.21.7), (19.21.8), (19.21.10), (19.25.7), (19.25.10) and (19.25.11) were written as products to ensure the correct multivalued behavior.

    Reported by Luc Maisonobe on 2021-06-07

  • Additions

    Equations: (3.3.3_1), (3.3.3_2), (5.15.9) (suggested by Calvin Khor on 2021-09-04), (8.15.2), Pochhammer symbol representation in (10.17.1) for a k ( ν ) coefficient, Pochhammer symbol representation in (11.9.4) for a k ( μ , ν ) coefficient, and (12.14.4_5).

  • Equation (11.11.1)

    Pochhammer symbol representations for the functions F k ( ν ) and G k ( ν ) were inserted.

  • Equation (35.7.3)

    Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument F 1 2 was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

  • Equations (15.6.1)–(15.6.9)

    The Olver hypergeometric function F ( a , b ; c ; z ) , previously omitted from the left-hand sides to make the formulas more concise, has been added. In Equations (15.6.1)–(15.6.5), (15.6.7)–(15.6.9), the constraint | ph ( 1 - z ) | < π has been added. In (15.6.6), the constraint | ph ( - z ) | < π has been added. In Section 15.6 Integral Representations, the sentence immediately following (15.6.9), “These representations are valid when | ph ( 1 - z ) | < π , except (15.6.6) which holds for | ph ( - z ) | < π .”, has been removed.

  • 10: Bibliography K
  • S. H. Khamis (1965) Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution. Justus von Liebig Verlag, Darmstadt (German, English).
  • A. I. Kheyfits (2004) Closed-form representations of the Lambert W function. Fract. Calc. Appl. Anal. 7 (2), pp. 177–190.