weight function

SearchAdvancedHelp

(0.002 seconds)

1—10 of 44 matching pages

1: 18.32 OP’s with Respect to Freud Weights

§18.32 OP’s with Respect to Freud Weights

►A Freud weight is a weight function of the form ►

18.32.1 $$w(x)=\exp \left(-Q(x)\right),$$ ,

ⓘ

Symbols:

$\exp z$: exponential function, $w(x)$: weight function and $x$: real variable

Referenced by:

§18.32

Permalink:

http://dlmf.nist.gov/18.32.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.32 and Ch.18

… ►For asymptotic approximations to OP’s that correspond to Freud weights with more general functions $Q(x)$ see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999). … ►

18.32.2 $$w(x)={\left|x\right|}^{\alpha }\exp \left(-Q(x)\right),$$ $x\in ℝ$, $\alpha >-1$,

ⓘ

Symbols:

$\in$: element of, $\exp z$: exponential function, $ℝ$: real line, $w(x)$: weight function and $x$: real variable

Referenced by:

§18.32, Erratum (V1.2.0) §18.32

Permalink:

http://dlmf.nist.gov/18.32.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.32 and Ch.18

…

2: 12.15 Generalized Parabolic Cylinder Functions

… ►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …

3: 18.31 Bernstein–Szegő Polynomials

… ►The Bernstein–Szegő polynomials $\{p_n(x)\}$, $n=0,1,\mathrm{\dots }$, are orthogonal on $(-1,1)$ with respect to three types of weight function: ${(1-x^2)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x^2)}^{\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x)}^{\frac{1}{2}}{(1+x)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$. …

4: 18.3 Definitions

… ►

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

► ►►

Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
…

►

… ►For $-1-\beta >\alpha >-1$ a finite system of Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ is orthogonal on $(1,\mathrm{\infty })$ with weight function $w(x)={(x-1)}^{\alpha }{(x+1)}^{\beta }$. …

5: 3.5 Quadrature

… ►An interpolatory quadrature rule … ►

Gauss Formula for a Logarithmic Weight Function

… ►

Table 3.5.17: Nodes and weights for the 20-point Gauss formula for the logarithmic weight function.

► ►►

$x_k$		$w_k$
…

►

… ►

Table 3.5.18: Nodes and weights for the 5-point complex Gauss quadrature formula with $s=1$.

► ►►

${\zeta }_k$				$w_k$
…

►

… ►

Table 3.5.21: Cubature formulas for disk and square.

► ►►

Diagram	$(x_j,y_j)$	$w_j$	$R$
…

►

6: 18.39 Applications in the Physical Sciences

… ►

§18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences

… ►Shizgal (2015, Chapter 2), contains a broad-ranged discussion of methods and applications for these, and other, non-classical weight functions. … ►

Table 18.39.1: Typical Non-Classical Weight Functions Of Use In DVR Applications^a

► ►►

Name of OP System	$w(x)$	$[a,b]$	Notation	Applications
…

►

… ►Graphs of the weight functions of (18.39.50) are shown in Figure 18.39.2. … ►In the attractive case (18.35.6_4) for the discrete parts of the weight function where with , are also simplified: …

7: 18.33 Polynomials Orthogonal on the Unit Circle

… ►A system of polynomials $\{{\varphi }_n(z)\}$, $n=0,1,\mathrm{\dots }$, where ${\varphi }_n(z)$ is of proper degree $n$, is orthonormal on the unit circle with respect to the weight function $w(z)$ ($\ge 0$) if … ►Instead of (18.33.9) one might take monic OP’s $\{q_n(x)\}$ with weight function $(1+x)w_1(x)$, and then express $q_n\left(\frac{1}{2}(z+z^{-1})\right)$ in terms of ${\varphi }_{2n}(z^{\pm 1})$ or ${\varphi }_{2n+1}(z^{\pm 1})$. … ►

18.33.19 $$d\mu (z)=\frac{1}{2\pi \mathrm{i}}w(z)\frac{dz}{z}$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $w(x)$: weight function and $z$: complex variable

Referenced by:

§18.33(vi)

Permalink:

http://dlmf.nist.gov/18.33.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33 and Ch.18

►for some weight function $w(z)$ ($\ge 0$) then (18.33.17) (see also (18.33.1)) takes the form … ►For $w(z)$ as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure $\mu$ in (18.33.17)) and with ${\alpha }_n$ the Verblunsky coefficients in (18.33.23), (18.33.24), Szegő’s theorem states that …

8: 18.2 General Orthogonal Polynomials

… ►A system (or set) of polynomials $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, where $p_n(x)$ has degree $n$ as in §18.1(i), is said to be orthogonal on $(a,b)$ with respect to the weight function $w(x)$ ($\ge 0$) if … ►For OP’s $\{p_n(x)\}$ on $ℝ$ with respect to an even weight function $w(x)$ we have … ►Under further conditions on the weight function there is an equiconvergence theorem, see Szegő (1975, Theorem 13.1.2). … ►

Monotonic Weight Functions

… ► …

9: 18.40 Methods of Computation

… ►

18.40.4 $$\underset{N\to \mathrm{\infty }}{lim}{F}_N(z)=F(z)\equiv \frac{1}{{\mu }_0}{\int }_a^b\frac{w(x)dx}{z-x},$$ $z\in ℂ\backslash [a,b]$,

ⓘ

Symbols:

$[a,b]$: closed interval, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\equiv$: equals by definition, $\int$: integral, $N$: positive integer, $w(x)$: weight function, $z$: complex variable and $x$: real variable

Proved:

Ismail (2009, p. 36)(proved)

Permalink:

http://dlmf.nist.gov/18.40.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

… ►

18.40.6 $$\underset{\epsilon \to 0+}{lim}{\int }_a^b\frac{w(x)dx}{x^{\prime }+\mathrm{i}\epsilon -x}dx={⨍}_a^b\frac{w(x)dx}{x^{\prime }-x}-\mathrm{i}\pi w(x^{\prime }),$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, ${⨍}_a^b$: Cauchy principal value, $w(x)$: weight function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.40.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

… ►

►►

Figure 18.40.1: Histogram approximations to the Repulsive Coulomb–Pollaczek, RCP, weight function integrated over $[-1,x)$, see Figure 18.39.2 for an exact result, for $Z=+1$, shown for $N=12$ and $N=120$. Magnify

… ►

18.40.8 $$w(x_{i,N})\approx \frac{w_{i,N}}{{\frac{dx(j,N)}{dj}|}_{j=i}}.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $(a,b)$: open interval, $N$: positive integer, $w(x)$: weight function, $w_x$: weights and $x$: real variable

Referenced by:

Figure 18.40.2, Figure 18.40.2

Permalink:

http://dlmf.nist.gov/18.40.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

… ►The example chosen is inversion from the ${\alpha }_n,{\beta }_n$ for the weight function for the repulsive Coulomb–Pollaczek, RCP, polynomials of (18.39.50). …

10: 18.19 Hahn Class: Definitions

… ►

Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, standardizations, and parameter constraints.

► ►►

	$p_n(x)$	$X$	$w_x$	$h_n$
…

►

… ►

18.19.2 $$w(z;a,b,\overline{a},\overline{b})=\mathrm{\Gamma }\left(a+iz\right)\mathrm{\Gamma }\left(b+iz\right)\mathrm{\Gamma }\left(\overline{a}-iz\right)\mathrm{\Gamma }\left(\overline{b}-iz\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\overline{z}$: complex conjugate, $w(x)$: weight function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/18.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

►

18.19.3 $$w(x)=w(x;a,b,\overline{a},\overline{b})={|\mathrm{\Gamma }\left(a+\mathrm{i}x\right)\mathrm{\Gamma }\left(b+\mathrm{i}x\right)|}^2,$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\overline{z}$: complex conjugate, $\mathrm{i}$: imaginary unit, $w(x)$: weight function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

… ►

18.19.7 $$w^{(\lambda )}(z;\varphi )=\mathrm{\Gamma }\left(\lambda +iz\right)\mathrm{\Gamma }\left(\lambda -iz\right){\mathrm{e}}^{(2\varphi -\pi )z},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $w(x)$: weight function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/18.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

…