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1: 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). …
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: 2.10 Sums and Sequences
Another version is the Abel–Plana formula: …
  • (c)

    The first infinite integral in (2.10.2) converges.

  • 4: 1.13 Differential Equations
    Then the following relation is known as Abel’s identity
    5: 1.2 Elementary Algebra
    §1.2(iv) Means
    6: Bibliography B
  • H. F. Baker (1995) Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.
  • 7: Errata
  • 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.