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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: 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). …
    7: Errata
  • Subsection 17.9(iii)

    The title of the paragraph which was previously “Gasper’s q -Analog of Clausen’s Formula” has been changed to “Gasper’s q -Analog of Clausen’s Formula (16.12.2)”.

  • Paragraph Inversion Formula (in §35.2)

    The wording was changed to make the integration variable more apparent.

  • Section 1.13

    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 ¶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.

  • Usability

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

  • Subsection 14.18(iii)

    This subsection now identifies Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

  • 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. …