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1: 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

… ►

1.18.12 $$⟨f,g⟩={\int }_a^{b}f(x)\overline{g(x)}d\alpha (x),$$

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Defines:

$⟨f,g⟩$: inner product over functions

Symbols:

$\overline{z}$: complex conjugate, $dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/1.18.E12

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

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

2: 1.2 Elementary Algebra

… ►

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

… ►

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

… ►

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

Encodings:

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

… ►

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

… ►

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

Encodings:

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

…

3: 1.1 Special Notation

… ► ►►►

$x,y$	real variables.
…
$⟨f,g⟩$	inner, or scalar, product for real or complex vectors or functions.
…

…

4: 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

…

5: 18.39 Applications in the Physical Sciences

… ►

$c_n=⟨\chi ,{\psi }_n⟩,$

…

6: 29.14 Orthogonality

… ►is orthogonal and complete with respect to the inner product … ►Each of the following seven systems is orthogonal and complete with respect to the inner product (29.14.2): …When combined, all eight systems (29.14.1) and (29.14.4)–(29.14.10) form an orthogonal and complete system with respect to the inner product …

7: Bibliography I

… ►

On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.

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External Links:

ISSN 0021-9045, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.36(ii).

Encodings:

BibTeX

…

8: 18.36 Miscellaneous Polynomials

… ►Sobolev OP’s are orthogonal with respect to an inner product involving derivatives. …

9: 31.15 Stieltjes Polynomials

… ►with respect to the inner product …

10: 26.10 Integer Partitions: Other Restrictions

… ►

26.10.2 $$\sum _{n=0}^{\mathrm{\infty }}p(𝒟,n)q^n=\prod _{j=1}^{\mathrm{\infty }}(1+q^j)=\prod _{j=1}^{\mathrm{\infty }}\frac{1}{1-q^{2j-1}}=1+\sum _{m=1}^{\mathrm{\infty }}\frac{q^{m(m+1)/2}}{(1-q)(1-q^2)\mathrm{\cdots }(1-q^m)}=1+\sum _{m=1}^{\mathrm{\infty }}{q}^m(1+q)(1+q^2)\mathrm{\cdots }(1+q^{m-1}),$$

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Symbols:

$p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

A&S Ref:

24.2.2

Permalink:

http://dlmf.nist.gov/26.10.E2

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Annotations for §26.10(ii), §26.10 and Ch.26

… ►

26.10.3 $$(1-x)\sum _{m,n=0}^{\mathrm{\infty }}{p}_m(\le k,𝒟,n)x^m{q}^n=\sum _{m=0}^k{\left[\genfrac{}{}{0.0pt}{}{k}{m}\right]}_q{q}^{m(m+1)/2}{x}^m=\prod _{j=1}^k(1+x{q}^j),$$ ,

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Referenced by:

§26.10(ii)

Permalink:

http://dlmf.nist.gov/26.10.E3

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Annotations for §26.10(ii), §26.10 and Ch.26

… ►

26.10.5 $$\sum _{n=0}^{\mathrm{\infty }}p(\in S,n)q^n=\prod _{j\in S}\frac{1}{1-q^j}.$$

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Symbols:

$\in$: element of, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $j$: nonnegative integer, $n$: nonnegative integer, $S$: set and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.10.E5

Encodings:

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Annotations for §26.10(ii), §26.10 and Ch.26

… ►where the inner sum is the sum of all positive odd divisors of $t$. … ►where the inner sum is the sum of all positive divisors of $t$ that are in $S$. …