DLMF: Untitled Document

… ►Mathematically, we scale the height to

h

lying in the interval

[0,4]

and the components are computed as follows … ►Specifically, by scaling the phase angle in

[0,2\pi)

to

q

in the interval

[0,4)

, the hue (in degrees) is computed as …

… ►

5.14.4

\int_{[0,1]^{n}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{2c}\prod_{k% =1}^{n}t_{k}^{a-1}(1-t_{k})^{b-1}\,\mathrm{d}t_{k}=\frac{1}{(\Gamma\left(1+c% \right))^{n}}\prod_{k=1}^{m}\frac{a+(n-k)c}{a+b+(2n-k-1)c}\*\prod_{k=1}^{n}% \frac{\Gamma\left(a+(n-k)c\right)\Gamma\left(b+(n-k)c\right)\Gamma\left(1+kc% \right)}{\Gamma\left(a+b+(2n-k-1)c\right)},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $[\NVar{a},\NVar{b}]$ : closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : nonnegative integer, $n$ : nonnegative integer, $k$ : nonnegative integer, $a$ : real or complex variable, $b$ : real or complex variable and $\Delta$ : product
Permalink:: http://dlmf.nist.gov/5.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §5.14, §5.14 and Ch.5

… ►

5.14.5

\int_{[0,\infty)^{n}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{2c}% \prod_{k=1}^{n}t_{k}^{a-1}e^{-t_{k}}\,\mathrm{d}t_{k}=\prod_{k=1}^{m}(a+(n-k)c% )\frac{\prod_{k=1}^{n}\Gamma\left(a+(n-k)c\right)\Gamma\left(1+kc\right)}{(% \Gamma\left(1+c\right))^{n}},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $[\NVar{a},\NVar{b})$ : half-closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $m$ : nonnegative integer, $n$ : nonnegative integer, $k$ : nonnegative integer, $a$ : real or complex variable and $\Delta$ : product
Permalink:: http://dlmf.nist.gov/5.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §5.14, §5.14 and Ch.5

… ►

5.14.6

\frac{1}{(2\pi)^{n/2}}\int_{(-\infty,\infty)^{n}}|\Delta(t_{1},\dots,t_{n})|^{% 2c}\prod_{k=1}^{n}\exp\left(-\tfrac{1}{2}t_{k}^{2}\right)\,\mathrm{d}t_{k}=% \frac{\prod_{k=1}^{n}\Gamma\left(1+kc\right)}{(\Gamma\left(1+c\right))^{n}},

\Re c>-1/n

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $\Re$ : real part, $n$ : nonnegative integer, $k$ : nonnegative integer and $\Delta$ : product
Referenced by:: §5.20
Permalink:: http://dlmf.nist.gov/5.14.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §5.14, §5.14 and Ch.5

… ►

5.14.7

\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}\prod_{1\leq j<k\leq n}|e^{i\theta_{j% }}-e^{i\theta_{k}}|^{2b}\,\mathrm{d}\theta_{1}\cdots\,\mathrm{d}\theta_{n}=% \frac{\Gamma\left(1+bn\right)}{(\Gamma\left(1+b\right))^{n}},

\Re b>-1/n

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $j$ : nonnegative integer, $n$ : nonnegative integer, $k$ : nonnegative integer and $b$ : real or complex variable
Referenced by:: §5.14, §5.20
Permalink:: http://dlmf.nist.gov/5.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §5.14, §5.14 and Ch.5

… ►

Orthogonality on Intervals

►Let

(a,b)

be a finite or infinite open interval in

\mathbb{R}

. … ►Assume that the interval

[a,b]

is bounded. … ►

The Nevai class ${\mathbf{M}}(a,b)$

… ►For OP’s

p_{n}

on

[a,b]

with weight function

w(x)

and orthogonality relation (18.2.5_5) assume that

b<\infty

and

w(x)

is non-decreasing in the interval

[a,b]

. …

… ►then there are exactly

\genfrac{(}{)}{0.0pt}{}{n+N-2}{N-2}

polynomials

S(z)

, each of which corresponds to each of the

\genfrac{(}{)}{0.0pt}{}{n+N-2}{N-2}

ways of distributing its

n

zeros among

N-1

intervals

(a_{j},a_{j+1})

,

j=1,2,\dots,N-1

. … ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index

\mathbf{m}=(m_{1},m_{2},\dots,m_{N-1})

, where each

m_{j}

is a nonnegative integer, there is a unique Stieltjes polynomial with

m_{j}

zeros in the open interval

(a_{j},a_{j+1})

for each

j=1,2,\dots,N-1

. … ►

31.15.8

S_{\mathbf{m}}(z_{1})S_{\mathbf{m}}(z_{2})\cdots S_{\mathbf{m}}(z_{N-1}),

z_{j}\in(a_{j},a_{j+1})

,

ⓘ

Symbols:: $\in$ : element of, $(\NVar{a},\NVar{b})$ : open interval, $z$ : complex variable, $j$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $S_{\mathbf{m}}(z)$ : Stieltjes polynomial
Referenced by:: §31.15(iii)
Permalink:: http://dlmf.nist.gov/31.15.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(iii), §31.15 and Ch.31

►

31.15.9

S_{\mathbf{l}}(z_{1})S_{\mathbf{l}}(z_{2})\cdots S_{\mathbf{l}}(z_{N-1}),

z_{j}\in(a_{j},a_{j+1})

,

ⓘ

Symbols:: $\in$ : element of, $(\NVar{a},\NVar{b})$ : open interval, $z$ : complex variable, $j$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $S_{\mathbf{m}}(z)$ : Stieltjes polynomial
Permalink:: http://dlmf.nist.gov/31.15.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(iii), §31.15 and Ch.31

… ►

31.15.10

Q=(a_{1},a_{2})\times(a_{2},a_{3})\times\cdots\times(a_{N-1},a_{N}),

ⓘ

Defines:: $Q$ : set (locally)
Symbols:: $(\NVar{a},\NVar{b})$ : open interval, $a$ : complex parameter and $N+1$ : number of singularities
Permalink:: http://dlmf.nist.gov/31.15.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(iii), §31.15 and Ch.31

…

… ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

► ►►

Name	$p_{n}(x)$	$(a,b)$	$w(x)$	$h_{n}$	$k_{n}$	$\ifrac{\tilde{k}_{n}}{k_{n}}$	Constraints
…

► … ►

Jacobi on Other Intervals

►For

-1-\beta>\alpha>-1

a finite system of Jacobi polynomials

P^{(\alpha,\beta)}_{n}\left(x\right)

is orthogonal on

(1,\infty)

with weight function

w(x)=(x-1)^{\alpha}(x+1)^{\beta}

. For

\nu\in\mathbb{R}

and

N>-\tfrac{1}{2}

a finite system of Jacobi polynomials

P^{(-N-1+\mathrm{i}\nu,-N-1-\mathrm{i}\nu)}_{n}\left(\mathrm{i}x\right)

(called pseudo Jacobi polynomials or Routh–Romanovski polynomials) is orthogonal on

(-\infty,\infty)

with

w(x)=\left(1+x^{2}\right)^{-N-1}{\mathrm{e}}^{2\nu\operatorname{arctan}x}

. …

… ► ►►►►

$\mathbb{L}$	lattice in $\mathbb{C}$ .
…
$[a,b]$ or $(a,b)$	closed, or open, straight-line segment joining $a$ and $b$ , whether or not $a$ and $b$ are real.
…
$G\times H$	Cartesian product of groups $G$ and $H$ , that is, the set of all pairs of elements $(g,h)$ with group operation $(g_{1},h_{1})+(g_{2},h_{2})=(g_{1}+g_{2},h_{1}+h_{2})$ .

…

… ►The negative zero

x_{-}(a)

decreases monotonically in the interval

-1<a<0

, and satisfies … ►

(a)

two zeros in each of the intervals $-2n<a<2-2n$ when $x<0$ ;

►

(b)

two zeros in each of the intervals $-2n<a<1-2n$ when $0<x\leq x_{n}^{*}$ ;

… ►

Table 8.13.1: Double zeros

(a_{n}^{*},x_{n}^{*})

of

\gamma^{*}\left(a,x\right)

.

► ►►

$n$	$a_{n}^{*}$		$x_{n}^{*}$
…

►

… ►Let

f(x)

be continuous on a closed interval

[a,b]

. … ►For general intervals

[a,b]

we rescale: … ►Let

f

be continuous on a closed interval

[a,b]

and

w

be a continuous nonvanishing function on

[a,b]

:

w

is called a weight function. …of type

[k,\ell]

to

f

on

[a,b]

minimizes the maximum value of

\left|\epsilon_{k,\ell}(x)\right|

on

[a,b]

, where … ►Two are endpoints:

(x_{0},y_{0})

and

(x_{3},y_{3})

; the other points

(x_{1},y_{1})

and

(x_{2},y_{2})

are control points. …

… ►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). The principal branches of

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

and

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

are real when

\nu

,

\mu\in\mathbb{R}

and

z\in(1,\infty)

. … ►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]

. …

… ►is computed with

p=1

on the interval

[0,30]

. … ►

[a,b]=[-1,1],

… ►

[a,b]=[-1,1],

… ►

[a,b]=[-1,1],

… ►

[a,b]=[0,1],

…

on intervals

21—30 of 237 matching pages

21: About Color Map

22: 5.14 Multidimensional Integrals

23: 18.2 General Orthogonal Polynomials

Orthogonality on Intervals

The Nevai class ${\mathbf{M}}(a,b)$

24: 31.15 Stieltjes Polynomials

25: 18.3 Definitions

Jacobi on Other Intervals

26: 23.1 Special Notation

27: 8.13 Zeros

28: 3.11 Approximation Techniques

29: 14.21 Definitions and Basic Properties

30: 3.5 Quadrature

on intervals

21—30 of 237 matching pages

21: About Color Map

22: 5.14 Multidimensional Integrals

23: 18.2 General Orthogonal Polynomials

Orthogonality on Intervals

The Nevai class 𝐌 ⁢ ( a , b )

24: 31.15 Stieltjes Polynomials

25: 18.3 Definitions

Jacobi on Other Intervals

26: 23.1 Special Notation

27: 8.13 Zeros

28: 3.11 Approximation Techniques

29: 14.21 Definitions and Basic Properties

30: 3.5 Quadrature

The Nevai class ${\mathbf{M}}(a,b)$