Abel–Plana formula
(0.002 seconds)
1—10 of 228 matching pages
1: 2.10 Sums and Sequences
§2.10(i) Euler–Maclaurin Formula
… ►This is the Euler–Maclaurin formula. Another version is the Abel–Plana formula: … ►The first infinite integral in (2.10.2) converges.
2: 1.15 Summability Methods
Abel Summability
… ►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
…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
5: 1.13 Differential Equations
6: Errata

•
In Paragraph Inversion Formula in §35.2, the wording was changed to make the integration variable more apparent.

•
In many cases, the links from mathematical symbols to their definitions were corrected or improved. These links were also enhanced with ‘tooltip’ feedback, where supported by the user’s browser.
 •

•
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 twoargument Wronskian
1.13.4$$\mathcal{W}\left\{{w}_{1}(z),{w}_{2}(z)\right\}=det\left[\begin{array}{cc}\hfill {w}_{1}(z)\hfill & \hfill {w}_{2}(z)\hfill \\ \hfill {w}_{1}^{\prime}(z)\hfill & \hfill {w}_{2}^{\prime}(z)\hfill \end{array}\right]={w}_{1}(z){w}_{2}^{\prime}(z){w}_{2}(z){w}_{1}^{\prime}(z)$$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 $\mathcal{W}\left\{{w}_{1}(z),\mathrm{\dots},{w}_{n}(z)\right\}=det\left[{w}_{k}^{(j1)}(z)\right]$, 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 $n$thorder 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 7542019 FloatingPoint Arithmetic Standard IEEE (2019). In the new standard, single, double and quad floatingpoint precisions are replaced with new standard names of binary32, binary64 and binary128. Figure 3.1.1 has been expanded to include the binary128 floatingpoint 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.

•
In Equations (15.6.1)–(15.6.9), the Olver hypergeometric function $\mathbf{F}(a,b;c;z)$, previously omitted from the lefthand sides to make the formulas more concise, has been added. In Equations (15.6.1)–(15.6.5), (15.6.7)–(15.6.9), the constraint $$ has been added. In (15.6.6), the constraint $$ has been added. In Section 15.6 Integral Representations, the sentence immediately following (15.6.9), “These representations are valid when $$, except (15.6.6) which holds for $$.”, has been removed.

•
In Subsection 25.2(ii) Other Infinite Series, it is now mentioned that (25.2.5), defines the Stieltjes constants ${\gamma}_{n}$. Consequently, ${\gamma}_{n}$ in (25.2.4), (25.6.12) are now identified as the Stieltjes constants.
 •

•
Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.
Previously this formula was expressed as an equality. Since this formula expresses an asymptotic expansion, it has been corrected by using instead an $\sim $ relation.
Reported by Gergő Nemes on 20190129
The formulas in these subsections are valid only for $x\ge 0$. No conditions on $x$ were given originally.
Reported 20101018 by Andreas Kurt Richter.