# 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
• Other Changes

• In Subsection 1.9(i), just below (1.9.1), a phrase was added which elaborates that ${\mathrm{i}}^{2}=-1$.

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

was added as an equality. In 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 $\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]$, 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.

• 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 ${{}_{2}F_{1}}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.