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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

\left\langle u,v\right\rangle\in\mathbb{C}

is defined for all

u,v\in V

with the properties: (i)

\left\langle u,v\right\rangle

is complex linear in

u

; (ii)

\left\langle u,v\right\rangle=\overline{\left\langle v,u\right\rangle}

; (iii)

\left\langle v,v\right\rangle\geq 0

; (iv) if

\left\langle v,v\right\rangle=0

then

v=0

. …Two elements

u

and

v

V

are orthogonal if

\left\langle u,v\right\rangle=0

. … ►Functions

f,g\in L^{2}\left(X,\,\mathrm{d}\alpha\right)

for which

\left\langle f-g,f-g\right\rangle=0

are identified with each other. … ►,

\left\langle u_{\lambda},u_{\lambda^{\prime}}\right\rangle=0

, for

\lambda\neq\lambda^{\prime}

. … ►The adjoint

{T}^{*}

T

does satisfy

\left\langle Tf,g\right\rangle=\left\langle f,{T}^{*}g\right\rangle

where

\left\langle f,g\right\rangle=\int_{a}^{b}f(x)g(x)\,\mathrm{d}x

. …

2: 1.2 Elementary Algebra

… ►

1.2.40

\left\langle\mathbf{u},\mathbf{v}\right\rangle=\sum_{i=1}^{n}u_{i}\overline{v_% {i}}={\mathbf{v}}^{{\rm H}}\mathbf{u}.

ⓘ

Defines:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors
Symbols:: ${\NVar{\mathbf{A}}}^{{\rm H}}$ : Hermitian conjugate of matrix, $\overline{\NVar{z}}$ : complex conjugate, $n$ : nonnegative integer, $\mathbf{v}$ : column vector and $\mathbf{u}$ : column vector
Referenced by:: §1.18(i), §1.2(v)
Permalink:: http://dlmf.nist.gov/1.2.E40
Encodings:: pMML, png, TeX
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►

1.2.41

\left\langle\mathbf{u},\mathbf{v}\right\rangle=\overline{\left\langle\mathbf{v% },\mathbf{u}\right\rangle},

ⓘ

Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\mathbf{v}$ : column vector and $\mathbf{u}$ : column vector
Permalink:: http://dlmf.nist.gov/1.2.E41
Encodings:: pMML, png, TeX
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►

1.2.42

\left\langle\alpha\mathbf{u},\beta\mathbf{v}\right\rangle=\alpha\overline{% \beta}\left\langle\mathbf{u},\mathbf{v}\right\rangle,

ⓘ

Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\mathbf{v}$ : column vector and $\mathbf{u}$ : column vector
Referenced by:: §1.2(v)
Permalink:: http://dlmf.nist.gov/1.2.E42
Encodings:: pMML, png, TeX
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►

1.2.43

\left\langle\mathbf{v},\mathbf{v}\right\rangle=0,

ⓘ

Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors and $\mathbf{v}$ : column vector
Permalink:: http://dlmf.nist.gov/1.2.E43
Encodings:: pMML, png, TeX
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►

1.2.44

\left\langle\mathbf{u},\mathbf{v}\right\rangle=0.

ⓘ

Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\mathbf{v}$ : column vector and $\mathbf{u}$ : column vector
Permalink:: http://dlmf.nist.gov/1.2.E44
Encodings:: pMML, png, TeX
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

…

… ►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the

icon) for links to defining formula. …

4: Karl Dilcher

… ►Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. …

5: 1.1 Special Notation

… ► ►►►

$x, y$	real variables.
…
$\left\langle f,g\right\rangle$	inner, or scalar, product for real or complex vectors or functions.
…

…

6: 27.1 Special Notation

… ► ►►►►►

$d, k, m, n$	positive integers (unless otherwise indicated).
…
$\sum_{d\mathbin{\|}n}$ , $\prod_{d\mathbin{\|}n}$	sum, product taken over divisors of $n$ .
$\sum_{\left(m,n\right)=1}$	sum taken over $m$ , $1\leq m\leq n$ and $m$ relatively prime to $n$ .
…
$\sum_{p}$ , $\prod_{p}$	sum, product extended over all primes.
…

7: Need Help?

… ►

Finding Things

•

How do I search within DLMF? See Guide to Searching the DLMF.
•

See also the Index or Notations sections.
•

Links to definitions, keywords, annotations and other interesting information can be found in the Info boxes by clicking or hovering the mouse over the icon next to each formula, table, figure, and section heading.

8: 11.12 Physical Applications

… ►Applications of Struve functions occur in water-wave and surface-wave problems (Hirata (1975) and Ahmadi and Widnall (1985)), unsteady aerodynamics (Shaw (1985) and Wehausen and Laitone (1960)), distribution of fluid pressure over a vibrating disk (McLachlan (1934)), resistive MHD instability theory (Paris and Sy (1983)), and optical diffraction (Levine and Schwinger (1948)). …

9: 15.18 Physical Applications

… ►The hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)). …

10: 19.38 Approximations

… ►They are valid over parts of the complex

k

and

\phi

planes. …