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1: 1.2 Elementary Algebra

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1.2.41 $$⟨𝐮,𝐯⟩=\overline{⟨𝐯,𝐮⟩},$$

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Symbols:

$\overline{z}$: complex conjugate, $⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector and $𝐮$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E41

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Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.42 $$⟨\alpha 𝐮,\beta 𝐯⟩=\alpha \overline{\beta }⟨𝐮,𝐯⟩,$$

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Symbols:

$\overline{z}$: complex conjugate, $⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector and $𝐮$: column vector

Referenced by:

§1.2(v)

Permalink:

http://dlmf.nist.gov/1.2.E42

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Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.43 $$⟨𝐯,𝐯⟩=0,$$

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Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors and $𝐯$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E43

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Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.44 $$⟨𝐮,𝐯⟩=0.$$

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Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector and $𝐮$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E44

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Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.46 $$\Vert 𝐯\Vert ={\Vert 𝐯\Vert }_2=\sqrt{⟨𝐯,𝐯⟩},$$

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Defines:

${\Vert 𝐯\Vert }_2$: $l$² norm and $\Vert 𝐯\Vert$: vector norm ($l$²)

Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors and $𝐯$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E46

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Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A complex linear vector space $V$ is called an inner product space if an inner product $⟨u,v⟩\in ℂ$ is defined for all $u,v\in V$ with the properties: (i) $⟨u,v⟩$ is complex linear in $u$; (ii) $⟨u,v⟩=\overline{⟨v,u⟩}$; (iii) $⟨v,v⟩\ge 0$; (iv) if $⟨v,v⟩=0$ then $v=0$. …Two elements $u$ and $v$ in $V$ are orthogonal if $⟨u,v⟩=0$. … ►

1.18.3 $$c_n=⟨v,v_n⟩.$$

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Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors and $n$: nonnegative integer

Referenced by:

§1.18(i)

Permalink:

http://dlmf.nist.gov/1.18.E3

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Annotations for §1.18(i), §1.18 and Ch.1

… ►The adjoint $T^{\ast }$ of $T$ does satisfy $⟨Tf,g⟩=⟨f,T^{\ast }g⟩$ where $⟨f,g⟩={\int }_a^{b}f(x)g(x)dx$. … ►

1.18.70 $$⟨Tv,w⟩=⟨v,Tw⟩,$$ $v,w\in 𝒟(T)$.

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Symbols:

$\in$: element of, $⟨𝐮,𝐯⟩$: inner product over vectors and $w$: variable

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http://dlmf.nist.gov/1.18.E70

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Annotations for §1.18(ix), §1.18(ix), §1.18 and Ch.1

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3: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The adjoint of a matrix $𝐀$ is the matrix ${𝐀}^{\ast }$ such that $⟨𝐀𝐚,𝐛⟩=⟨𝐚,{𝐀}^{\ast }𝐛⟩$ for all $𝐚,𝐛\in {𝐄}_n$. … ►

1.3.20 $$𝐮=\sum _{i=1}^n{c}_i{𝐚}_i,$$ $c_i=⟨𝐮,{𝐚}_i⟩$.

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Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors and $n$: nonnegative integer

Referenced by:

Erratum (V1.2.0) Section 1.3

Permalink:

http://dlmf.nist.gov/1.3.E20

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Annotations for §1.3(iv), §1.3(iv), §1.3 and Ch.1

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4: 1.6 Vectors and Vector-Valued Functions

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Dot Product (or Scalar Product)

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Cross Product (or Vector Product)

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1.6.9

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Symbols:

$det$: determinant, $\sin z$: sine function, $\Vert 𝐯\Vert$: vector norm ($l$²), $\theta$: angle between $𝐚$ and $𝐛$ , $𝐢$: unit vector, $𝐣$: unit vector and $𝐤$: unit vector

Referenced by:

§1.6(i)

Permalink:

http://dlmf.nist.gov/1.6.E9

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

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Figure 1.6.1: Vector notation. … Magnify

… ►Much vector algebra involves summation over suffices of products of vector components. …

5: 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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6: 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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7: 21.6 Products

§21.6 Products

… ►Two such vectors are considered equivalent if their difference is a vector with integer elements. …where ${𝐜}_j$ and ${𝐝}_j$ are arbitrary $h$-dimensional vectors. … ►Then …Thus $𝝂$ is a $g$-dimensional vector whose entries are either $0$ or $1$. …

8: Bibliography D

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Vector coupling coefficients as products of prime factors. Comput. Phys. Comm. 4 (2), pp. 268–274.

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Notes:

Single-precision Fortran

External Links:

Document, Code

Referenced by:

4th item.

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9: 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). …

10: 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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