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1: 1.1 Special Notation

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$x,y$	real variables.
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$⟨f,g⟩$	inner, or scalar, product for real or complex vectors or functions.
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2: 21.1 Special Notation

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$g,h$	positive integers.
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$𝐚\cdot 𝐛$	scalar product of the vectors $𝐚$ and $𝐛$.
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3: 1.6 Vectors and Vector-Valued Functions

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Dot Product (or Scalar Product)

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1.6.2 $$𝐚\cdot 𝐛=a_1{b}_1+a_2{b}_2+a_3{b}_3.$$

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Referenced by:

§1.2(v)

Permalink:

http://dlmf.nist.gov/1.6.E2

Encodings:

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See also:

Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1

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4: 1.2 Elementary Algebra

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§1.2(v) Matrices, Vectors, Scalar Products, and Norms

… ►Column vectors $𝐮$ and $𝐯$ of the same length $n$ have a scalar product … ►The scalar product has properties …

5: 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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6: 31.17 Physical Applications

… ►The problem of adding three quantum spins $𝐬$, $𝐭$, and $𝐮$ can be solved by the method of separation of variables, and the solution is given in terms of a product of two Heun functions. We use vector notation $[𝐬,𝐭,𝐮]$ (respective scalar $(s,t,u)$) for any one of the three spin operators (respective spin values). …

7: Bibliography

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SPHEREPACK 2.0: A Model Development Facility. NCAR Technical Note Technical Report TN-436-STR, National Center for Atmospheric Research.

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Notes:

SPHEREPACK 2.0 is a collection of Fortran programs that facilitates computer modeling of geophysical processes. Accurate solutions are obtained via the spectral method that uses both scalar and vector spherical harmonic transforms. The package also contains utility programs for computing the associated Legendre functions. SPHEREPACK 3.1 was released in August 2003.

External Links:

FullText, SPHEREPACK 3.1

Referenced by:

1st item.

Encodings:

BibTeX

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Integrals of products of Bernoulli polynomials. J. Math. Anal. Appl. 381 (1), pp. 10–16.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Pablo Manuel Román), ZentralBlatt

Referenced by:

§24.13(i), §24.13(i).

Encodings:

BibTeX

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Integrals of products of Airy functions. J. Phys. A 10 (4), pp. 485–490.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

(9.11.10), (9.11.5), (9.11.6), (9.11.7), (9.11.8), (9.11.9), §9.11(iv), §9.11(iv).

Encodings:

BibTeX

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An integral of products of ultraspherical functions and a $q$-extension. J. London Math. Soc. (2) 33 (1), pp. 133–148.

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External Links:

ISSN 0024-6107, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§10.22(iv), §14.30(ii).

Encodings:

BibTeX

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8: 3.3 Interpolation

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${\mathrm{\ell }}_k(z)=\underset{j=0\phantom{\prime }}{\overset{n\phantom{\prime }}{{\displaystyle {\prod }^{\prime }}}}{\displaystyle \frac{z-z_j}{z_k-z_j}},$

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3.3.3 $${\omega }_{n+1}(z)=\prod _{k=0}^n(z-z_k),$$

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Defines:

${\omega }_{n+1}(z)$: nodal polynomial (locally)

Symbols:

$z_k$: nodes of function

Referenced by:

§3.3(iii), Erratum (V1.1.3) for Subsection 3.3(i)

Permalink:

http://dlmf.nist.gov/3.3.E3

Encodings:

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See also:

Annotations for §3.3(i), §3.3 and Ch.3

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3.3.3_1 $${\tilde{\omega }}_k=\frac{1}{{\omega }_{n+1}^{\prime }(z_k)}=\underset{j=0\phantom{\prime }}{\overset{n\phantom{\prime }}{{\prod }^{\prime }}}\frac{1}{z_k-z_j}.$$

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Symbols:

$z_k$: nodes of function and ${\omega }_{n+1}(z)$: nodal polynomial

Referenced by:

§3.3(i), Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/3.3.E3_1

Encodings:

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Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §3.3(i), §3.3 and Ch.3

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3.3.12 $$c_n=\frac{1}{(n+1)!}\max \prod _{k=n_0}^{n_1}\left|t-k\right|,$$

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Defines:

$c_n$: bound coefficient (locally)

Symbols:

$!$: factorial (as in $n!$)

Referenced by:

§3.4(i)

Permalink:

http://dlmf.nist.gov/3.3.E12

Encodings:

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See also:

Annotations for §3.3(ii), §3.3 and Ch.3

… ►It can be used for solving a nonlinear scalar equation $f(z)=0$ approximately. …