Abel

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1: 22.18 Mathematical Applications

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§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

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Abel Summability

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Abel Means

… ► $A(r,\theta )$ is a harmonic function in polar coordinates (1.9.27), and … ►Here $u(x,y)=A(r,\theta )$ is the Abel (or Poisson) sum of $f(\theta )$, and $v(x,y)$ has the series representation … ►

Abel Summability

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3: 2.10 Sums and Sequences

… ►Another version is the Abel–Plana formula: … ►

The first infinite integral in (2.10.2) converges.

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4: 1.13 Differential Equations

… ►Then the following relation is known as Abel’s identity …

5: Bibliography B

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Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.

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Notes:

Reprint of the 1897 original, with a foreword by Igor Krichever.

External Links:

ISBN 0-521-49877-5, MathReview (John B. Little), ZentralBlatt

Referenced by:

§20.9(ii).

Encodings:

BibTeX

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6: 1.2 Elementary Algebra

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§1.2(iv) Means

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7: Errata

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In Equation (1.13.4), the determinant form of the two-argument Wronskian

1.13.4

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 $𝒲\left\{w_1(z),\mathrm{\dots },w_n(z)\right\}=det\left[w_k^{(j-1)}(z)\right]$, where $1\le j,k\le n$. 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.

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