# Abel

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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 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 …
##### 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
##### 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
• Section 1.13

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

1.13.4 $\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=\det\begin{bmatrix}w_{1}(z)&w_{2}(% z)\\ w_{1}^{\prime}(z)&w_{2}^{\prime}(z)\end{bmatrix}=w_{1}(z)w_{2}^{\prime}(z)-w_{% 2}(z)w_{1}^{\prime}(z)$

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 $\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]$, where $1\leq j,k\leq n$. 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.