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Abel–Plana formula

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1: 2.10 Sums and Sequences
§2.10(i) Euler–Maclaurin Formula
This is the Euler–Maclaurin formula. Another version is the AbelPlana formula: …
  • (c)

    The first infinite integral in (2.10.2) converges.

  • 2: 1.15 Summability Methods
    Abel Summability
    Abel Means
    A ( r , θ ) is a harmonic function in polar coordinates ((1.9.27)), and … Here u ( x , y ) = A ( r , θ ) is the Abel (or Poisson) sum of f ( θ ) , and v ( x , y ) has the series representation …
    Abel Summability
    3: 22.18 Mathematical Applications
    §22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem
    For any two points ( x 1 , y 1 ) and ( x 2 , y 2 ) on this curve, their sum ( x 3 , y 3 ) , always a third point on the curve, is defined by the Jacobi–Abel addition law …a construction due to Abel; see Whittaker and Watson (1927, pp. 442, 496–497). …
    4: Bibliography B
  • H. F. Baker (1995) Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.
  • W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
  • B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.
  • B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.
  • R. Bo and R. Wong (1999) A uniform asymptotic formula for orthogonal polynomials associated with exp ( - x 4 ) . J. Approx. Theory 98, pp. 146–166.
  • 5: 1.13 Differential Equations
    Then the following relation is known as Abel’s identity
    6: Errata
  • Changes


    • In Paragraph Inversion Formula in §35.2, the wording was changed to make the integration variable more apparent.

    • In many cases, the links from mathematical symbols to their definitions were corrected or improved. These links were also enhanced with ‘tooltip’ feedback, where supported by the user’s browser.

  • Other Changes


    • In Subsection 1.9(i), just below (1.9.1), a phrase was added which elaborates that i 2 = - 1 .

    • Poor spacing in math was corrected in several chapters.

    • In Section 1.13, there were several modifications. In Equation (1.13.4), the determinant form of the two-argument Wronskian

      1.13.4
      𝒲 { w 1 ( z ) , w 2 ( z ) } = det [ w 1 ( z ) w 2 ( z ) w 1 ( z ) w 2 ( z ) ] = w 1 ( z ) w 2 ( z ) - w 2 ( z ) w 1 ( z )

      was added as an equality. In Paragraph Wronskian in §1.13(i), immediately below Equation (1.13.4), a sentence was added indicating that in general the n -argument Wronskian is given by 𝒲 { w 1 ( z ) , , w n ( z ) } = det [ w k ( j - 1 ) ( z ) ] , where 1 j , k n . Immediately below Equation (1.13.4), a sentence was added giving the definition of the n -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for n th-order differential equations. A reference to Ince (1926, §5.2) was added.

    • In Section 3.1, there were several modifications. In Paragraph IEEE Standard in §3.1(i), the description was modified to reflect the most recent IEEE 754-2019 Floating-Point Arithmetic Standard IEEE (2019). In the new standard, single, double and quad floating-point precisions are replaced with new standard names of binary32, binary64 and binary128. Figure 3.1.1 has been expanded to include the binary128 floating-point memory positions and the caption has been updated using the terminology of the 2019 standard. A sentence at the end of Subsection 3.1(ii) has been added referring readers to the IEEE Standards for Interval Arithmetic IEEE (2015, 2018). This was suggested by Nicola Torracca.

    • In 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.

  • Other Changes


    • In 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.

    • In Subsection 25.2(ii) Other Infinite Series, it is now mentioned that (25.2.5), defines the Stieltjes constants γ n . Consequently, γ n in (25.2.4), (25.6.12) are now identified as the Stieltjes constants.

    • In (25.11.36) we have emphasized the link with the Dirichlet L -function, and used the fact that χ ( k ) = 0 . A sentence just below (25.11.36) was added indicating that one should make a comparison with (25.15.1) and (25.15.3).

    • Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.

  • Equation (33.11.1)


    33.11.1
    H ± ( η , ρ ) e ± i θ ( η , ρ ) k = 0 ( a ) k ( b ) k k ! ( ± 2 i ρ ) k

    Previously this formula was expressed as an equality. Since this formula expresses an asymptotic expansion, it has been corrected by using instead an relation.

    Reported by Gergő Nemes on 2019-01-29

  • Subsections 1.15(vi) and 1.15(vii)


    The formulas in these subsections are valid only for x 0 . No conditions on x were given originally.

    Reported 2010-10-18 by Andreas Kurt Richter.

  • 7: 27.20 Methods of Computation: Other Number-Theoretic Functions
    The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function p ( n ) for n < N . … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
    8: 1.2 Elementary Algebra
    §1.2(iv) Means
    9: Howard S. Cohl
    Howard is the project leader for the NIST Digital Repository of Mathematical Formulae seeding and development projects. In this regard, he has been exploring mathematical knowledge management and the digital expression of mostly unambiguous context-free full semantic information for mathematical formulae.
    10: Preface
    Abramowitz and Stegun’s Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables is being completely rewritten with regard to the needs of today. …The authors will review the relevant published literature and produce approximately twice the number of formulas that were contained in the original Handbook. …