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31—40 of 151 matching pages

31: 26.15 Permutations: Matrix Notation

… ►For

(j,k)\in B

B\setminus[j,k]

denotes

B

after removal of all elements of the form

(j,t)

(t,k)

t=1,2,\ldots,n

. … ►

26.15.5

R(x,B)=x\,R(x,B\setminus[j,k])+R(x,B\setminus(j,k)).

ⓘ

Symbols:: $[\NVar{a},\NVar{b}]$ : closed interval, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $x$ : real variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $B$ : subset and $R(x,B)$ : rook polynomial
Permalink:: http://dlmf.nist.gov/26.15.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §26.15 and Ch.26

…

32: 1.5 Calculus of Two or More Variables

… ►Suppose that

a, b, c

are finite,

d

is finite or

+\infty

, and

f(x,y)

\ifrac{\partial f}{\partial x}

are continuous on the partly-closed rectangle or infinite strip

[a,b]\times[c,d)

. …for all

c_{1}\in[c_{0},d)

and all

x\in[a,b]

. … ►Let

f(x,y)

be defined on a closed rectangle

R=[a,b]\times[c,d]

. For …let

(\xi_{j},\eta_{k})

denote any point in the rectangle

[x_{j},x_{j+1}]\times[y_{k},y_{k+1}]

j=0,\dots,n-1

k=0,\dots,m-1

. …

33: 1.10 Functions of a Complex Variable

… ►Suppose the subarc

z(t)

t\in[t_{j-1},t_{j}]

is contained in a domain

D_{j}

j=1,\dots,n

. … ►Let

C

be a simple closed contour consisting of a segment

\mathit{AB}

of the real axis and a contour in the upper half-plane joining the ends of

\mathit{AB}

. … ►Let

D

be a domain and

[a,b]

be a closed finite segment of the real axis. Assume that for each

t\in[a,b]

f(z,t)

is an analytic function of

z

D

, and also that

f(z,t)

is a continuous function of both variables. … ►For each

t\in[a,b)

f(z,t)

is analytic in

D

;

f(z,t)

is a continuous function of both variables when

z\in D

and

t\in[a,b)

; the integral (1.10.18) converges at

b

, and this convergence is uniform with respect to

z

in every compact subset

S

D

. …

34: 1.8 Fourier Series

… ►where

f(x)

is square-integrable on

[-\pi,\pi]

and

a_{n},b_{n},c_{n}

are given by (1.8.2), (1.8.4). … ►For

f(x)

piecewise continuous on

[a,b]

and real

\lambda

, … ►If

a_{n}

and

b_{n}

are the Fourier coefficients of a piecewise continuous function

f(x)

[0,2\pi]

, then … ►Suppose that

f(x)

is continuous and of bounded variation on

[0,\infty)

. Suppose also that

f(x)

is integrable on

[0,\infty)

and

f(x)\to 0

x\to\infty

. …

35: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Let

X=[a,b]

[a,b)

(a,b]

(a,b)

be a (possibly infinite, or semi-infinite) interval in

\mathbb{R}

. … ►Consider the second order differential operator acting on real functions of

x

in the finite interval

[a,b]\subset\mathbb{R}

… ►The nature of these extensions for unbounded intervals such as

[0,\infty)

, and unbounded operators on them, are the subject of §1.18(ix). … ►

Example 1: Bessel–Hankel Transform, $X=[0,\infty)$

… ►For

f(x)

piecewise continuously differentiable on

[0,\infty)

…

36: 3.5 Quadrature

… ►where

h=b-a

f\in C^{2}[a,b]

, and

a<\xi<b

. … ►Let

h=\frac{1}{2}(b-a)

and

f\in C^{4}[a,b]

. … ►If

f\in C^{2m+2}[a,b]

, then the remainder

E_{n}(f)

in (3.5.2) can be expanded in the form … ►is computed with

p=1

on the interval

[0,30]

. … ►Rules of closed type include the Newton–Cotes formulas such as the trapezoidal rules and Simpson’s rule. …

37: 14.21 Definitions and Basic Properties

… ►When

z

is complex

P^{\pm\mu}_{\nu}\left(z\right)

Q^{\mu}_{\nu}\left(z\right)

, and

\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)

are defined by (14.3.6)–(14.3.10) with

x

replaced by

z

: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when

z\in(1,\infty)

, and by continuity elsewhere in the

z

-plane with a cut along the interval

(-\infty,1]

; compare §4.2(i). … ►Many of the properties stated in preceding sections extend immediately from the

x

-interval

(1,\infty)

to the cut

z

-plane

\mathbb{C}\backslash(-\infty,1]

. …

38: 25.12 Polylogarithms

… ►The principal branch has a cut along the interval

[1,\infty)

and agrees with (25.12.1) when

|z|\leq 1

; see also §4.2(i). … ►

25.12.3

\operatorname{Li}_{2}\left(z\right)+\operatorname{Li}_{2}\left(\frac{z}{z-1}% \right)=-\frac{1}{2}(\ln\left(1-z\right))^{2},

z\in\mathbb{C}\setminus[1,\infty)

ⓘ

Symbols:: $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\in$ : element of, $\ln\NVar{z}$ : principal branch of logarithm function, $\setminus$ : set subtraction, $x$ : real variable and $z$ : complex variable
Keywords:: connection formula
Source:: Maximon (2003, (3.4), p. 2808)
A&S Ref:: 27.7.5 (is a modified form with $z=1-x$ )
Permalink:: http://dlmf.nist.gov/25.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §25.12(i), §25.12 and Ch.25

►

25.12.4

\operatorname{Li}_{2}\left(z\right)+\operatorname{Li}_{2}\left(\frac{1}{z}% \right)=-\frac{1}{6}\pi^{2}-\frac{1}{2}(\ln\left(-z\right))^{2},

z\in\mathbb{C}\setminus[0,\infty)

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\in$ : element of, $\ln\NVar{z}$ : principal branch of logarithm function, $\setminus$ : set subtraction and $z$ : complex variable
Keywords:: connection formula
Source:: Maximon (2003, (3.2), p. 2808)
Referenced by:: §25.12(ii)
Permalink:: http://dlmf.nist.gov/25.12.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §25.12(i), §25.12 and Ch.25

…

39: 1.13 Differential Equations

… ►

§1.13(vii) Closed-Form Solutions

… ►on a finite interval

[a,b]\subset\mathbb{R}

, this is then a regular Sturm-Liouville system. … ►Equation (1.13.26) with

x\in[a,b]

may be transformed to the Liouville normal form ►

1.13.29

\ddot{w}(t)+\left(\lambda-\widehat{q}(t)\right)w(t)=0,

t\in[0,c]

ⓘ

Symbols:: $[\NVar{a},\NVar{b}]$ : closed interval, $\in$ : element of, $\widehat{q}(t)$ : function, $w(t)$ : solution, $\lambda$ : real parameter, $t$ : real variable, $c$ : real variable and $\ddot{\NVar{z}}$ : 2nd derivative of $\NVar{w}$ with respect to $t$
Referenced by:: §1.13(viii), §1.18
Permalink:: http://dlmf.nist.gov/1.13.E29
Encodings:: pMML, png, TeX
See also:: Annotations for §1.13(viii), §1.13(viii), §1.13 and Ch.1

… ►As the interval

[a,b]

is mapped, one-to-one, onto

[0,c]

by the above definition of

t

, the integrand being positive, the inverse of this same transformation allows

\widehat{q}(t)

to be calculated from

p,q,\rho

in (1.13.31),

p,\rho\in C^{2}(a,b)

and

q\in C(a,b)

. …

40: 21.1 Special Notation

… ► ►►►►

$g, h$	positive integers.
…
$\boldsymbol{{\alpha}},\boldsymbol{{\beta}}$	$g$ -dimensional vectors, with all elements in $[0,1)$ , unless stated otherwise.
…
$a\circ b$	intersection index of $a$ and $b$ , two cycles lying on a closed surface. $a\circ b=0$ if $a$ and $b$ do not intersect. Otherwise $a\circ b$ gets an additive contribution from every intersection point. This contribution is $1$ if the basis of the tangent vectors of the $a$ and $b$ cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is $-1$ .
…

…

closed

31—40 of 151 matching pages

31: 26.15 Permutations: Matrix Notation

32: 1.5 Calculus of Two or More Variables

33: 1.10 Functions of a Complex Variable

34: 1.8 Fourier Series

35: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

Example 1: Bessel–Hankel Transform, X = [ 0 , ∞ )

36: 3.5 Quadrature

37: 14.21 Definitions and Basic Properties

38: 25.12 Polylogarithms

39: 1.13 Differential Equations

§1.13(vii) Closed-Form Solutions

40: 21.1 Special Notation

Example 1: Bessel–Hankel Transform, $X=[0,\infty)$