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31: 3.5 Quadrature

… ►The nodes $x_1,x_2,\mathrm{\dots },x_n$ are prescribed, and the weights $w_k$ and error term $E_n(f)$ are found by integrating the product of the Lagrange interpolation polynomial of degree $n-1$ and $w(x)$. … ►

§3.5(x) Cubature Formulas

… ►and the square $S$, given by $\left|x\right|\le h$, $\left|y\right|\le h$: … ►

Table 3.5.21: Cubature formulas for disk and square.

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Diagram	$(x_j,y_j)$	$w_j$	$R$
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32: Bibliography K

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Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution. Justus von Liebig Verlag, Darmstadt (German, English).

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Notes:

In cooperation with W. Rudert

External Links:

MathReview, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Jacobi polynomials. II. An analytic proof of the product formula. SIAM J. Math. Anal. 5, pp. 125–137.

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External Links:

Document, MathReview (R. P. Soni), ZentralBlatt

Referenced by:

§18.12, §18.17(ii), §18.18(vi).

Encodings:

BibTeX

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33: 3.4 Differentiation

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3.4.3 $$h{R}_{n,t}^{\prime }=\frac{h^{n+1}}{(n+1)!}\left(f^{(n+1)}({\xi }_0)\frac{d}{dt}\prod _{k=n_0}^{n_1}(t-k)+f^{(n+2)}({\xi }_1)\prod _{k=n_0}^{n_1}(t-k)\right),$$

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Defines:

$R_{n,t}^{\prime }(x)$: remainder (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/3.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(i), §3.4 and Ch.3

… ►For additional formulas involving values of ${\nabla }^{2}u$ and ${\nabla }^{4}u$ on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546). …

34: 14.30 Spherical and Spheroidal Harmonics

… ►For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii). … ►Here, in spherical coordinates, $L^2$ is the squared angular momentum operator: …

35: 10.9 Integral Representations

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§10.9(iii) Products

… ►where the square root has its principal value. … ► …

36: 18.18 Sums

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Expansion of $L^2$ functions

►In all three cases of Jacobi, Laguerre and Hermite, if $f(x)$ is $L^2$ on the corresponding interval with respect to the corresponding weight function and if $a_n,b_n,d_n$ are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in $L^2$ sense. … ►For integral representations for products implied by (18.18.8) and (18.18.9) see (18.17.5) and (18.17.6), respectively. …

37: Bibliography B

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Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.12.

Encodings:

BibTeX

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Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-360-9, MathReview (Hongyuan Zha), ZentralBlatt

Referenced by:

§3.11(v).

Encodings:

BibTeX

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38: Errata

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The following additions were made in Chapter 1:

• New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

• The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).

• In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).

• In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

• New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

• New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

• Another form of Parseval’s formula, (1.14.7_5).

• We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

• Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

• An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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Factors inside square roots on the right-hand sides of formulas (19.18.6), (19.20.10), (19.20.19), (19.21.7), (19.21.8), (19.21.10), (19.25.7), (19.25.10) and (19.25.11) were written as products to ensure the correct multivalued behavior.

Reported by Luc Maisonobe on 2021-06-07

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