logarithmic weight function

SearchAdvancedHelp

(0.004 seconds)

11—20 of 23 matching pages

11: 18.14 Inequalities

… ►

18.14.8 $${\mathrm{e}}^{-\frac{1}{2}x}\left|L_n^{(\alpha )}\left(x\right)\right|\le L_n^{(\alpha )}\left(0\right)=\frac{{\left(\alpha +1\right)}_n}{n!},$$ , $\alpha \ge 0$.

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Proved:

Koornwinder (1977, Remark 4.1)(proved)

Proof sketch:

The case $\alpha =0$ follows from the result on monotonic weight functions in §18.2(xii).

A&S Ref:

22.14.13

Referenced by:

§18.14(i), §18.2(xii)

Permalink:

http://dlmf.nist.gov/18.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

…

12: 18.28 Askey–Wilson Class

… ►

18.28.3 $$2\pi \sin \theta w(\cos \theta )={\left|\frac{{({\mathrm{e}}^{2i\theta };q)}_{\mathrm{\infty }}}{{(a{\mathrm{e}}^{\mathrm{i}\theta },b{\mathrm{e}}^{\mathrm{i}\theta },c{\mathrm{e}}^{\mathrm{i}\theta },d{\mathrm{e}}^{\mathrm{i}\theta };q)}_{\mathrm{\infty }}}\right|}^2,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $\sin z$: sine function, $w(x)$: weight function and $q$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

…

13: 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
…

►

… ►

18.3.2 $$x_{N+1,n}=\cos \left((n-\frac{1}{2})\pi /(N+1)\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.3, §18.3 and Ch.18

… ►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). … ►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 $w(x)={(x-1)}^{\alpha }{(x+1)}^{\beta }$. …

14: 1.7 Inequalities

… ►

1.7.8 $$\min (a_1,a_2,\mathrm{\dots },a_n)\le M(r)\le \max (a_1,a_2,\mathrm{\dots },a_n),$$

ⓘ

Symbols:

$n$: nonnegative integer and $M(r)$: weighted mean

A&S Ref:

3.2.2

Permalink:

http://dlmf.nist.gov/1.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §1.7(iii), §1.7 and Ch.1

… ►

1.7.9 $$M(r)\le M(s),$$ ,

ⓘ

Symbols:

$M(r)$: weighted mean

A&S Ref:

3.2.4

Permalink:

http://dlmf.nist.gov/1.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.7(iii), §1.7 and Ch.1

… ►For $f$ integrable on $[0,1]$, , and $\varphi$ convex on $(a,b)$ (§1.4(viii)), … ►

1.7.11

ⓘ

Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/1.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §1.7(iv), §1.7 and Ch.1

►For $\exp$ and $\ln$ see §4.2.

15: 1.4 Calculus of One Variable

… ►

§1.4(i) Monotonicity

… ► … ►For the function $\ln$ see §4.2(i). … ►For $\alpha (x)$ nondecreasing on the closure $I$ of an interval $(a,b)$, the measure $d\alpha$ is absolutely continuous if $\alpha (x)$ is continuous and there exists a weight function $w(x)\ge 0$, Riemann (or Lebesgue) integrable on finite subintervals of $I$, such that … ►

§1.4(viii) Convex Functions

…

16: 18.18 Sums

… ►

§18.18(i) Series Expansions of Arbitrary Functions

… ►

Expansion of $L^2$ functions

►In all three cases of Jacobi, Laguerre and Hermite, if $f(x)$ is $L^2$ on the corresponding interval with respect to the corresponding weight function and if $a_n,b_n,d_n$ are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in $L^2$ sense. … ►

Laguerre

… ►For the modified Bessel function $I_{\nu }\left(z\right)$ see §10.25(ii). …

17: 3.11 Approximation Techniques

… ►

§3.11(iii) Minimax Rational Approximations

… ►Then the minimax (or best uniform) rational approximation … ► $w(x)$ being a given positive weight function, and again $J\ge n+1$. Then (3.11.29) is replaced by … ► …

18: 18.38 Mathematical Applications

… ►If the nodes in a quadrature formula with a positive weight function are chosen to be the zeros of the $n$th degree OP with the same weight function, and the interval of orthogonality is the same as the integration range, then the weights in the quadrature formula can be chosen in such a way that the formula is exact for all polynomials of degree not exceeding $2n-1$. … ►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. …Each of these typically require a particular non-classical weight functions and analysis of the corresponding OP’s. … ►Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. … ►

Non-Classical Weight Functions

…

19: Bibliography K

… ►

Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity. J. Comput. Appl. Math. 205 (1), pp. 186–206.

ⓘ

External Links:

ISSN 0377-0427, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§19.12, §19.12, §19.36(iv), §19.36(iv).

Encodings:

BibTeX

… ►

Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

ⓘ

External Links:

ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)

Referenced by:

§18.32.

Encodings:

BibTeX

… ►

Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument. Comput. Phys. Comm. 4, pp. 221–226.

ⓘ

Notes:

Single-precision Fortran, accuracy 12-14S

External Links:

Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Orthogonal polynomials with weight function ${(1-x)}^{\alpha }{(1+x)}^{\beta }+M\delta (x+1)+N\delta (x-1)$ . Canad. Math. Bull. 27 (2), pp. 205–214.

ⓘ

External Links:

ISSN 0008-4395, MathReview (Wolfgang Hahn), ZentralBlatt

Referenced by:

§18.36(i), §18.36(i).

Encodings:

BibTeX

… ►

Strong asymptotics of polynomials orthogonal with respect to Freud weights. Internat. Math. Res. Notices 1999 (6), pp. 299–333.

ⓘ

External Links:

ISSN 1073-7928, Document, MathReview (Heinrich Begehr), ZentralBlatt

Referenced by:

§18.32.

Encodings:

BibTeX

…

20: 31.9 Orthogonality

… ►

§31.9(i) Single Orthogonality

… ►The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning. … ►For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64). ►

§31.9(ii) Double Orthogonality

… ►

31.9.6 $$\rho (s,t)=(s-t){(st)}^{\gamma -1}{\left((s-1)(t-1)\right)}^{\delta -1}{\left((s-a)(t-a)\right)}^{ϵ-1},$$

ⓘ

Defines:

$\rho (s,t)$: weight function (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter and $a$: complex parameter

Permalink:

http://dlmf.nist.gov/31.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.9(ii), §31.9 and Ch.31

…